“Pay what you want” as a profitable pricing strategy:
Theory and experimental evidence
Vincent Mak
[email protected], Judge Business School, University of Cambridge, Trumpington Street,
Cambridge CB2 1AG, United Kingdom
Rami Zwick
[email protected], Department of Management and Marketing, A. Gary Anderson Graduate School of Management, University of California, Riverside, Riverside CA 92521
Akshay R. Rao
[email protected], Department of Marketing & Logistics Management, Carlson School of Management, University of Minnesota,
May 8, 2010
The authors would like to acknowledge the comments of Tony Cui, George John, Amnon Rapoport, as well as seminar participants at the University of California, Berkeley, the University of California, Riverside, and the University of Cambridge, on earlier versions of this manuscript.
“Pay what you want” as a profitable pricing strategy:
Theory and experimental evidence
Abstract
Prevailing wisdom in the literature suggests that the success of a “pay what you want” (PWYW)
pricing strategy depends on consumers’ altruistic inclinations, sense of fair play, or consumers’
willingness to reciprocate the firm’s generous offer. In this article, we consider whether PWYW
can be profitable when the consumer’s only motive is self-interest. Through analyzing an
infinitely repeated pricing game between a firm and a fixed consumer population, we find that
PWYW could be as profitable as a fixed price regime, if the firm’s strategy is to threaten to
switch to the fixed price regime over a length of time in the future, should PWYW not yield
enough profits. If consumers are forward looking and there exist a sufficient number of
consumers who find the firm’s fixed price regime to be more costly than other alternatives, then
such a threat could lead to a sustainable PWYW regime. We also examine coordination
mechanisms that may enhance the sustainability of this regime in practice, and present a
laboratory study that provides support for some of our conclusions.
Keywords: Pay what you want; pay as you wish; participative pricing; experimental economics;
game theory.
1
1. Introduction
In late 2007, the British band Radiohead launched their new album In Rainbows with an unusual
strategy: customers could download the album from a specially created website and pay any
price they wanted. If they so desired, they could pay nothing. This “pay what you want”
(PWYW) campaign ran from October 10 to December 10, 2007. It generated considerable
commentary in the music industry, including skepticism regarding its financial viability1. For, if
consumers were allowed to download the new album for free, wouldn’t they all do just that?
Indeed, it was later estimated that about 62% of downloads of In Rainbows in October 2007 were
free downloads, but a remarkable 38% of downloads were accompanied by a payment that
averaged roughly $6.002 (Elberse and Bergsman 2008). Since there was no revenue sharing with
record companies and retailers, the entire revenue accrued to the band, leading one band member
to observe that in terms of digital income they “made more money out of this record than out of
all the other Radiohead albums put together” (Thom Yorke, quoted in Wired magazine, issue
16.01, January 2008).
More recently (October 2009), to celebrate their one year anniversary, the developers of the
indie video game “World of Goo” ran an experiment for one week, during which buyers could
pay what they wished for the game whose list price was $20. During the first week of the
experiment, about 57,000 people downloaded the game from the website and the average price
paid for the game was $2.03. The developers declared the results a “huge success” and extended
1 Fortune Magazine listed Radiohead’s PWYW strategy among the “101 Dumbest Moments in Business” in 2007 (http://money.cnn.com/galleries/2007/fortune/0712/gallery.101_dumbest.fortune/59.html). 2 The average payment is a misleading statistic in this case. Jane Dyball, Warner Chappell’s Head of Business Affairs, the publishing company that licensed all digital rights on behalf of Radiohead, pointed out that the band and their management never announced a timeline for their PWYW experiment and were watching the average price daily with a view to potentially withdrawing it any moment should it drop too low. Dyball pointed out that the average price went down after the download moved from uberfans to less committed fans, as expected (http://musically.com/blog/2008/10/15/exclusive-warner-chappell-reveals-radioheads-in-rainbows-pot-of-gold/).
2
the experiment for another week3. The success declaration should be evaluated with respect to
the reported 90% piracy rate for the game. Following the success of “World of Goo”, the
developer of Crayon Physics Deluxe allowed consumers to pay as they wished for the game for
one week (January 8-15, 2010). During this week 31,332 downloaded the game and paid on
average $1.90, leading the developer to announce that the experiment was “way more popular”
than he expected4.
Such PWYW pricing strategies are neither new nor rare, and are prevalent in a variety of
contexts. Public radio and television stations in the United States employ membership drives for
donations that contain all the essential features of a PWYW strategy. Many museums do not
charge admission fees, but suggest to patrons that they donate whatever they want, and
restaurants in the U.S., U.K., Australia and Spain have experimented with PWYW options on
their menus (Gregory 2009). As a consequence, PWYW has recently begun to attract thoughtful
academic scrutiny (e.g. Kim et al. 2009, Regner and Barria 2009, Gautier and van der Klaauw
2010, Gneezy et al. 2010, Chen et al. 2010). Many arguments for the successful employment of
a PWYW strategy can be proffered. One class of arguments relies on revenue generation
through consumers’ “social preferences”, which include their sense of altruism and fair play as
well as their willingness to reciprocate the firm’s generous offer. Another argument is based on
the notion of “loss leadership” and the generation of secondary income through an increased
customer base and through cross selling. Musicians, for example, can gain publicity by giving
away albums, which then leads to enhanced concert ticket sales and merchandizing; museums
may generate traffic through free admissions and subsequently earn revenues through sales at
their gift shops and cafeterias. However, such derivative income may not always be essential for
3 The developers, 2D Boy, posted the results of the experiment at http://2dboy.com/2009/10/26/pay-what-you-want-birthday-sale-wrap-up/ 4 We thank Petri Purho, the developer of Crayon Physics Deluxe, for sharing the sales data with us.
3
PWYW to be successful. Consider Wikipedia’s attempts to generate donations at the end of 2008,
when a banner with a “Donate Now” button appeared at the top of every Wikipedia page; the
campaign eventually achieved its target of raising $6 million, and there was no secondary
income associated with the strategy (Hughes 2009).
Our study examines whether PWYW strategies can successfully be employed when
arguments based on social preferences and loss leadership are not operative. Specifically, if a
firm is faced with purely self-interested consumers, and there are no secondary benefits, can
PWYW still be a profitable pricing strategy relative to fixed pricing (FP)? We answer this
question through a game-theoretic model as well as laboratory experimentation, and identify
conditions under which PWYW is more beneficial than FP to both the firm and to consumers.
The principal intuition underlying our results is captured in Wikipedia’s 2008 fundraising
campaign. In that campaign, Wikipedia underscored the argument that user donations “will help
keep Wikipedia free”, and hinted that a failure to raise sufficient funds would lead to Wikipedia
users being charged a subscription fee, or Wikipedia being forced to carry advertisements
(Hughes 2009). In essence, sufficient donations would sustain Wikipedia’s PWYW model in the
future, but if donations did not reach sufficiency, Wikipedia might begin to charge users a
subscription fee that would potentially be higher than the donation solicited, or Wikipedia might
entertain advertisements, an action that could reduce users’ future benefits because Wikipedia
might no longer be truly neutral.5
In our model, we abstract the essential features of the strategy just described and investigate
when it is incentive compatible for consumers. Consumers may be willing to pay a positive
5 The indie music download site Marathon of Dope, which operates on a PWYW basis, voices a plea of a similar nature in its web pages (e.g. http://marathonofdope.com/?p=634): “Whatever money you decide is reasonable to pay for the music at Marathon Of Dope will go directly to the artists. That money will enable them to continue making high caliber music and bring it directly to you.” The implicit threat to listeners is that insufficient current donations will force the artists to compromise on quality and efficient distribution of their music in the future.
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amount under PWYW if the alternative they face is a relatively high fixed price. That is, if the
firm’s willingness to switch to a high fixed price for a pre-specified length of time (which we
call “punishment periods”) is a credible threat, then consumers have an incentive to pay a
positive amount in the current period. This argument stands if every individual consumer’s time-
discounted total expected utility gain under PWYW is at least as high as the corresponding total
utility gain that she would enjoy if she paid nothing in the current period under PWYW and then
paid fixed prices throughout the punishment periods. Our major theoretical results consist of a
statement of this intuition in quantitative terms. Formally, we examine a class of simple firm
strategies in the following form in the context of infinitely repeated firm-consumers interactions:
(1) the firm continues to implement PWYW if total payment from consumers is not lower than a
pre-announced threshold; (2) if the threshold is not reached in a particular period, the firm
implements FP for a pre-specified number of periods from the next period onwards, before
reverting to PWYW. Given this firm strategy, we find conditions under which a PWYW-
sustaining equilibrium exists on the consumers’ side, which generates higher profit than under
FP. In other words, we identify parametric conditions that admit a PWYW equilibrium which is
more profitable to the firm and more beneficial to consumers, than if the firm practices FP.
However, our equilibrium solutions are sensitive to coordination failures, because, given a
threshold profit and a punishment strategy that fulfill the theoretical conditions, there is typically
a great multitude of consumer-side equilibria, which makes it potentially difficult for consumers
to align their behavior with respect to one particular equilibrium. Thus, we suggest four
mechanisms that can be adopted by the firm to enhance consumer coordination. The functions of
two of these mechanisms – payment constraints and probabilistic punishment – can be
demonstrated purely through game theoretic considerations. The other two, namely that
5
consumers are allowed to communicate with each other, and that consumers are provided
suggested payments, are behavioral strategies inspired by previous literature, and require
empirical verification. Therefore we report a laboratory experiment designed to understand: (a)
whether (i) consumer communication and (ii) suggested payment enable subjects to sustain a
theoretically admissible PWYW equilibrium; (b) as control conditions, whether (i) theoretically
admissible PWYW equilibria are sustainable without coordination mechanisms, and (ii) whether
PWYW is sustainable even when theory does not admit any PWYW equilibria. We conclude
our paper with a discussion of the theoretical insights we obtain and their relevance to
practitioners.
2. Literature and Model
In this section, we first offer a brief review of the literature that is relevant to the PWYW
phenomenon. We then present an analytical model that captures our core argument that self-
interested consumers might, under certain conditions, sustain a PWYW strategy that is profitable
for the firm.
2.1 Related Literature
While the literature on pricing strategies is voluminous, the literature that speaks to the PWYW
pricing strategy is relatively sparse. One approach examines the secondary benefits that might
accrue to the firm if PWYW strategies are employed. For instance, the viability of PWYW in the
context of museum admissions has been studied empirically by Steiner (1997) and Toepler
(2006). Steiner (1997) examines whether free admission can serve as a loss leader that increases
overall museum revenues through souvenir sales and cafeteria sales, but finds empirically that
this pricing strategy is not optimal. Toepler (2006) finds that visitors to a museum with free
6
admission typically do not donate as much as the amount recommended by the museum
establishment. Seemingly, the recommended amount serves as a reference point, and consumers
discount from that reference point (Rao and Sieben 1992).
A number of recent field studies on PWYW, including Kim et al. (2009), Regner and Barria
(2009), Gautier and van der Klaauw (2010), and Gneezy et al. (2010) find that consumers often
pay positive amounts, and identify factors that affect their payments. Kim et al. (2009) observe
that social preferences such as altruism and a sense of fair play influence the amounts paid.
Regner and Barria (2009) suggest that consumers’ motivation to reciprocate the firm’s offer to
allow comprehensive pre-purchase sampling (if there is such an offer) influences their PWYW
payment positively. Gautier and van der Klaauw (2009), meanwhile, demonstrate that
consumers who were told they could pay what they wanted after committing to a purchase on
average paid more than those who were told they could pay what they wanted before committing
to a purchase.
Our paper complements this line of empirical research by examining PWYW when a firm’s
credible threat of a high future fixed price provides the sole motivation for self-interested
consumers to pay a positive amount under PWYW. The mechanism we propose involves the
continued offer of PWYW by the firm in the subsequent period, subject to the requirement that
the sum of consumer payments in the current period reaches a specific threshold. This
requirement makes our model similar to the provision point public goods game that has been
studied experimentally (e.g. van der Kragt et al. 1983, Rapoport 1988, Cadsby and Maynes 1998,
and Croson and Marks 2000). That is, our study points out how an infinitely repeated pricing
game involving PWYW can produce incentives for consumers to pay sufficiently high prices
under PWYW, through a mechanism that resembles a provision point public goods game.
7
However, whereas our mechanism is about consumers paying to avoid the discontinuation of a
beneficial pricing regime, the provision point public goods game is about players contributing to
facilitate the provision of a beneficial public good. This variation in framing might make a
behavioral difference. Further, to tackle the issue of potential coordination failures in our model,
we also suggest additional coordination mechanisms, some of which are not discussed in the
provision point public goods literature.
PWYW can also be seen as a type of participative pricing mechanism (Chandran and
Morwitz 2005), in which consumers are involved in the price-setting process. But common
participative pricing mechanisms such as price negotiations, conventional auctions, and the
Name-Your-Own-Price (NYOP) mechanism (e.g. Kamins et al. 2004, Terwiesch et al. 2005,
Spann and Tellis 2006, Amaldoss and Jain 2008), are qualitatively different from PWYW
because, while a seller can reject a price bid in those other participative pricing mechanisms, he
cannot do so when practicing PWYW.
2.2 Model
Consider an infinitely repeated game6 in which a firm S launches and sells a perishable product
at zero marginal cost in every period to a fixed market environment (marginal cost is indeed
negligible with Internet businesses and with museums, among which there are many examples of
PWYW).7 The products have no intrinsic value to the firm, and, being perishable, a product sold
in a period has no utility to consumers once that period is over.
6 An “infinitely repeated” game can also be interpreted as a repeated game with indefinite termination. That is, conditioned on the game being played now, there is always a positive probability that there will be a next period in which the game is played again (see, e.g. Zwick et al. 1992). The time discount factors of players in an infinitely repeated game partly capture this probability (and partly capture their intrinsic time preferences) in an expected utility framework. 7 Adding a per period fixed cost does not alter the essence of our results, and hence we assume in this study that the per period fixed cost is zero unless stated otherwise (but see Appendix B where we consider the implications of a positive marginal cost).
8
We assume that every consumer in any one period demands at most one unit of the available
alternatives.8 Firm S’s potential market consists of two principal consumer segments: (1) a
segment of dedicated fans (e.g., “Deadheads”) whose valuation of every one of S’s products is
always uF (> 0) and for whom there is no outside option that is a viable substitute, and (2) a
segment of casuals whose valuation of every one of S’s product is always uC (> 0) but for whom
there exists an outside option with the same valuation uC that is a perfect substitute, and this
outside option is sold at an exogenous, fixed price p0. Other than these two segments, there may
also exist some consumers who strictly prefer their best alternatives over S’s products even when
the latter are offered free, and therefore, they are not included in S’s potential market and can
safely be ignored. It is commonly known that the total number of fans and casuals is N, of which
a proportion q are fans and a proportion 1-q are casuals. Table 1 lists the valuations of the
different segments for the different available options in every period.
---------------------------------------
Insert Table 1 about here
---------------------------------------
At the beginning of every period, S decides whether to allow PWYW or to implement FP for
the product sold in that period. It may choose one and only one pricing scheme from this set of
two alternatives. If it decides to implement FP, it decides the price p and announces p to
consumers. Each consumer then decides whether to buy the product from the firm at price p, to
buy the outside option at price p0, or to not buy. If, instead, S decides to allow PWYW, it
announces this decision to consumers at the beginning of the period. Each consumer then
8 This is not a crucial assumption. As long as we specify consumers’ preference structures correctly, we can derive similar conditions for profitable PWYW even when allowing for multiunit per period demand. But no major additional insights are obtained by relaxing this assumption. Therefore, for expository convenience, we adopt the single-unit demand assumption.
9
decides whether to buy the product from S (and if so, to pay a non-negative price of her choice),
to buy the outside option at price p0, or to not buy. Regardless of the pricing scheme employed,
every consumer’s purchase decisions are independent of the purchase decisions of other
consumers. Figure 1 is a schematic representation of the sequence of decisions in each period.
We assume that, at the end of every period, S knows the total profit it earns during that period.9
---------------------------------------
Insert Figure 1 about here
---------------------------------------
All consumers have the same per period time discount factor )1,0[∈δ for utility gains.10 For
any consumer, the net utility gain of a purchase t periods after the current period is )(1 rut −−δ ,
where u is the consumer’s valuation of the purchased product and r is the price paid for that
purchase. Assume without loss of generality that 0puC ≥ ; if this is not true, the casuals’
effective outside option becomes no purchase. Assume also that 0puF ≥ .11
In most of our discussion, we assume that N, q, uF , uC, δ, and p0 are the same in all periods,
so that consumer taste and the marketing environment are stable and fixed across periods.
(However, as we will show later, some of these assumptions can be relaxed if the firm imposes
payment constraints under PWYW.) Our assumptions approximate some naturally occurring
scenarios. For instance, a popular music band with a well-established style often faces a largely
stable market consisting of many “casual” consumers with generic taste plus a small segment of
9 For the equilibria to be discussed, we do not need to assume that S knows the breakdown of individual payments under PWYW. 10 S may time-discount future profits too, but this turns out to be irrelevant for the equilibria we examine and hence, in the interest of brevity, we omit discussion of this aspect of the model unless otherwise stated. 11 Relaxing this assumption introduces the possibility of uF < p0 ≤ uC, so that, under FP, S might like to sell to casuals to the exclusion of its fans, whose valuations of S’s products are lower than the valuations of casuals towards S’s products. Accommodating this possibility does not affect the basic insights that follow from Proposition 1, but we adopt this assumption for expository convenience.
10
dedicated “fans”. Such a band has almost zero marginal cost when it offers a downloadable
album every season. It is also fairly typical with pop music that an album has virtually no value
to many consumers once a season is over, and thus can be viewed as a perishable product. The
outside competitive environment can be the mainstream music industry, which, despite the
constant change in stars and albums, presents more or less the same fare (as far as consumption
utility and price are concerned) to both fans and casuals.
It is helpful to first consider a firm strategy with which S implements FP with the same price
every period. Define p̂ as the price that maximizes S’s per period profit. The value of p̂ is
obtained as follow, together with the corresponding consumer purchase behavior, which
constitutes a demand-side equilibrium given the firm’s strategy:
(a) If 0pquF > , then Fup =ˆ ; only fans buy from S with zero net payoffs, so that S earns a
profit of FNqu , while every casual buys the outside option with net utility gain 0puC − ;
(b) If 0pquF < , then 0ˆ pp = ; every consumer buys from S with net utility gain 0pu − ,
where u is the consumer’s valuation of S’s product. S earns a profit of 0Np ;
(c) If 0pquF = , then either Fup =ˆ and consumer behavior is like (a), or 0ˆ pp = and
consumer behavior is like (b). S earns a profit of 0NpNquF = in either case.
We have made the convenient assumption that, if a consumer is indifferent between buying from
S and either buying the outside option or not buying at all, she buys from S.12 S’s per period
profit under this equilibrium is },max{ 00 pquN F=π .
Next, define the following firm strategy, which we call a “PWYW strategy with threshold π
and m punishment periods”: S announces a threshold profit level, π , that it must earn in a period
12 This assumption is not crucial. Relaxing it leads to some adjustment in our results but the major insights are unaltered.
11
in which it allows PWYW, if it is to continue to allow PWYW in the next period. In other words,
if total payment to S in a period with PWYW falls below π , S implements FP with price p̂ for m
periods (m > 0) from the next period onwards.
We now examine consumer-side equilibria that are admitted by such a firm strategy. First, it
is straightforward to see that, as long as Fu≥π (which must be true in the interesting case of
≥π },max{ 00 pquN F=π ), it is a consumer-side equilibrium that every consumer pays nothing
whenever S implements PWYW, thus triggering FP at price p̂ in the next m periods, during each
of which, fans buy from S while casuals buy from S if p̂ is lower than 0p or buy the outside
option otherwise. Call this equilibrium the baseline equilibrium. The baseline equilibrium is
feasible under a PWYW strategy whenever ≥π 0π regardless of the number of punishment
periods. Note that S can earn more profit by implementing FP perpetually, if it foresees that
consumers would behave according to the baseline equilibrium under PWYW.
However, there may also exist consumer-side equilibria under a PWYW strategy in which
the punishment is never triggered and S implements PWYW in every period. We shall focus on
equilibria with stationary, pure strategies, so that consumer i always pays the same amount ip to
S in any period with PWYW; such consumer strategies can be represented by the set }{ ip , which
we call a payment scheme. The total payment in a consumer-side equilibrium that sustains
PWYW cannot exceedπ ;13 since it cannot be lower than π for S to continue to implement
PWYW, total payment must be exactlyπ . This implies that π=∑i
ip is a PWYW-sustaining
13 Total payment cannot be higher than the threshold in equilibrium. Suppose this is not true and there is a payment scheme in equilibrium with a total payment that is higher than the threshold. But then there must exist an alternative payment scheme that also adds up to the threshold but requires at least one consumer to pay less while all other consumers pay the same. That consumer would then have an incentive to deviate unilaterally from the original payment scheme, meaning that the original payment scheme cannot be part of the equilibrium.
12
consumer-side equilibrium. Naturally, from the perspective of the firm, only strategies that yield
per period profits that are at least 0π are of interest. The question that then emerges is: under
what conditions will a PWYW strategy with threshold ≥π 0π and m punishment periods admit
PWYW-sustaining consumer-side equilibria? Our answer is summarized as follows, which is
proved in Appendix A:
PROPOSITION 1. A PWYW strategy with threshold ≥π 0π and m punishment periods
admits consumer-side equilibria in which S always allows PWYW, iff:
(a) 0pquF ≥ , and
(b) 1)1(
11
)1( 01 ≥
−+⋅
−−
+F
m
m
qupq
δδδ ,
and π satisfies:
FFmm quNpqqu ≥≥−+−− + /])1([)]1/()1([ 0
1 πδδδ .
Further, for every such π , any member of this set is characterized by a payment scheme }{ ip
with which consumer i pays ip for S’s product in any period with PWYW, and which satisfies:
(i) Fmm
i up )]1/()1([0 1+−−≤≤ δδδ for every i who is a fan,
(ii) 01 )]1/()1([0 pp mm
i+−−≤≤ δδδ for every i who is a casual, and
(iii) Fi
i Nqup ≥=∑ π .
Conceptually, in a consumer-side equilibrium that sustains PWYW, consumers are willing to
pay positive amounts because they reckon that paying nothing in a period, only to have to suffer
from the FP punishment in the next m periods, does not yield as much time-discounted total
utility as adhering to the payment scheme }{ ip every period. The conditions in Proposition 1
13
describe when this can happen at a profit threshold that is attractive to the firm relative to
perpetual FP. Further, these equilibria are Pareto superior to perpetual FP for both the firm and
consumers. When condition (a) is fulfilled, Fup =ˆ , FNqu=0π , so that S earns no less than 0π
per period with a PWYW strategy that fully satisfies the proposition. This outcome occurs
because, when S implements FP every period with price Fup =ˆ , only fans buy from S while
casuals buy the outside option, while under a PWYW equilibrium, all consumers buy from S and,
since S’s market size has now expanded from qN to N, S can still earn at least as much under
PWYW as under perpetual FP, even though every consumer pays less under PWYW. Individual
consumers who pay positive amounts in equilibrium are prevented from unilaterally deviating
and paying less than under equilibrium because of S’s threat that, if it earns a lower-than-
equilibrium profit in any period, the punishment of m periods of FP at price p̂ will be triggered.
The different upper bounds for payment under PWYW for the different types of consumers
indicate that the consumer-side equilibria bear some similarity to second-degree price
discrimination mechanisms, according to which consumers with different valuations of the target
product self-select the prices they pay.
Conditions (a) and (b) in Proposition 1 allow us to identify potential consumer markets that
can sustain a PWYW consumer-side equilibrium that provides more profit to the firm than under
perpetual FP. In Appendix A, we show that these conditions are in fact necessary for a PWYW
strategy with m punishment periods to admit any pure strategy PWYW-sustaining consumer-side
equilibria that generate at least as much per period profit as 0π . Condition (a) implies that a
portion of consumers do not buy from S under perpetual FP. Condition (b) is satisfied when: (1)
consumers’ time discount factor is high (δ is high); (2) the maximum profit that S may ever earn
from the casuals (which is realized when it charges a fixed price 0p ), namely 0)1( pqN − , is not
14
too much below the maximum profit that it may ever earn from the fans i.e., FNqu . Note that, if
condition (a) is fulfilled, we must have Fqu > 0)1( pq− and also 00 )1()( pqpuq F −≥− , which
means that the loss of profit from fans under FP when the price is reduced from Fu to 0p cannot
be compensated for by the corresponding gain of profit from casuals. But condition (b) is
satisfied when this difference is small and δ is high. To sum up, a consumer market admits a
PWYW sustaining consumer-side equilibrium when consumers are very forward looking and a
portion of them are “priced out” under FP only because a fixed price at Fu captures slightly
more profit than at 0p (in Appendix B, we describe a generalization of our model that allows for
a high degree of heterogeneity among consumers and a positive marginal cost).
We now turn to a discussion of two major issues underlying our theoretical results, before
presenting the empirical portion of our research which is designed to examine the effectiveness
of the two behavioral coordination mechanisms that we shall introduce in the discussion.
2.3 Discussion
Two key issues merit discussion here: the credibility of the firm’s threat to revert to a fixed price
regime if the payments from PWYW are insufficient, and how coordination failures on the part
of consumers might be avoided.
Credibility of the Firm’s Threat. It is important to ask if the firm’s “commitment to punish”
is credible from the point of view of consumers; if not, no PWYW equilibria can be realized.
Indeed, one might question whether, after a period in which PWYW is allowed but total payment
fails to reach threshold, S would prefer to “let bygones be bygones” and re-establish a PWYW
equilibrium. The firm may have an incentive to do this because, if a PWYW equilibrium can
successfully be re-established with a threshold that is higher than 0π , it can earn higher profits
by doing so, compared with the profits from implementing the punishment. Consumers could
15
prefer to sustain a PWYW equilibrium anew over the FP punishment because they pay less under
the PWYW regime. In other words, both the firm and consumers may agree to re-establish a
PWYW regime since it is Pareto superior to the firm implementing the punishment. Demanding
“renegotiation-proofness” rules out many commonly known repeated game equilibria, such as
the infinite repetition of (defect, defect) or the trigger strategy (cooperate, cooperate) equilibria
in repeated prisoners’ dilemma (see e.g. Farrell and Maskin 1989, Abreu et al. 1993).
However, the lack of renegotiation-proofness is not always a valid theoretical reason to rule
out an equilibrium (e.g., Fudenberg and Tirole, 1996, p.174-176), nor is it a legitimate reason to
discard a mechanism as unworkable. First of all, firms may find it preferable to make decisions
about the future based on consumers’ behavior in the past, compared with letting “bygones be
bygones”. That is, firms may legitimately be concerned that they may develop a reputation for
being “soft” on executing punishment, and consumers may rationally anticipate that should
PWYW not yield sufficiently high revenues during the next iteration, the firm will once again let
bygones be bygones.
In our case, if the firm believes that, following a period in which consumers had failed to pay
sufficiently under PWYW, there is a sufficiently high probability that consumers would change
their behavior and collectively pay to sustain PWYW at the firm’s desired threshold π upon re-
establishment of PWYW, renegotiation is appropriate. But, such a belief is as open to challenge
as the opposite belief by the firm that consumers are likely to fail to sustain PWYW if the firm
does not commit to a punishment. Both beliefs are “rational” although only the latter is
consistent with the PWYW equilibrium we suggest. Further, if the firm carries out punishment
as promised, a PWYW equilibrium is indeed “re-established” after m periods when the
punishment is over. The firm has to earn per period profit at 0π for m periods, but in exchange,
16
it establishes credibility among consumers regarding its threat when PWYW is re-established at
a higher threshold profit than 0π . That is, we suggest that renegotiation can at most be a transient
issue (if at all), and the credibility of the firm’s threat among consumers can be established at
least in the long run.
From a more empirical point of view, consumers in general believe in the credibility of firms’
announced commitment to various pricing policies. One example is the no-haggling policy
practiced by (among others) car retailer CarMax and the manufacturers of automobile models
Saturn and Scion. The retailer Filene’s Basement famously employ pre-announced pricing
schemes (Bell and Starr 1994) which consumers believe it will carry out. Such policies may not
survive a theoretical renegotiation-proofness scrutiny but are nevertheless perceived as credible
by consumers.
Coordination mechanisms. When PWYW-sustaining consumer-side equilibria with ≥π
0π and m punishment periods do exist, there is no unique equilibrium. Typically, a great
multitude of PWYW equilibria, each corresponding to a feasible payment scheme, can satisfy the
constraints in Proposition 1. In addition, the baseline equilibrium (in which everyone pays zero
whenever S allows PWYW, and FP is implemented subsequently) is always feasible whenever
≥π 0π , even though it is not Pareto optimal when PWYW equilibria exist. The implication is
that consumers might fail to coordinate to sustain PWYW not because each consumer fails to
react “correctly”, but because either: (1) consumer assumptions and behavior are misaligned as
to which particular equilibrium to follow, which results in payments that do not add up to
threshold, or (2) one or more consumers make “irrational”, perhaps unintentional, errors (such as
simple calculation mistakes). Behaviorally, the issue of coordination failure is not unique to our
model but is shared with all other multiple equilibrium games (e.g., Zwick and Rapoport 2002).
17
It can be envisioned that, given a PWYW firm strategy, over a time horizon that is much
longer than m, consumers can learn to reduce unintentional errors and can coordinate so as to
converge towards an equilibrium. Nevertheless, they may converge to the baseline equilibrium
rather than to any PWYW equilibrium. Therefore, it would be to the firm’s advantage to design
a mechanism that would help consumers align their behavior according to a particular PWYW
equilibrium in the very first place, and help them maintain such an alignment. We offer four
such possibilities:
1. Payment constraints. Suppose S only allows payments under PWYW to be one of three levels:
0, k, and l; moreover, S chooses k, l, and threshold profitπ , such that
Fmmmm ulpk )]1/()1([)]1/()1([0 1
01 ++ −−≤<−−≤< δδδδδδ and kqNNql )1( −+=π .
Then the unique PWYW equilibrium (other than the baseline equilibrium) has every fan (casual)
paying l (k). This occurs because a fan (casual) will pay at most l (k) in any PWYW equilibrium
according to the upper bounds presented in the latter part of Proposition 1 (which still apply
when there are payment constraints), so that total payment reaches π under payment constraints
only when every fan (casual) pays l (k). Payment constraints can therefore trim down the
number of PWYW equilibria to one, which greatly facilitates coordination. Moreover, if the
firm or consumers have uncertainty over the current and/or future values ofδ , Fu , or 0p , but it is
commonly known that:
(a) Fmm u)]1/()1([ 1+−− δδδ and 0
1 )]1/()1([ pmm +−− δδδ will never be below l and k
respectively, and
18
(b) kqNNql )1( −+=π will never be less than the optimal time-discounted average profit for S
under perpetual FP starting from any period,14 then the firm will maintain a threshold π during
every period while every fan (casual) paying l (k) remains an PWYW equilibrium payment
scheme, irrespective of the uncertainties.
It is arguable that payment constraints compromise the spirit of PWYW by allowing
consumers to pay only at certain levels, but notice that paying zero remains a real option, and, in
principle, every consumer still has a choice to decide how much to pay for S’s product. By
allowing zero payment, however, the baseline equilibrium (in which everyone pays zero
whenever S allows PWYW) remains feasible. The firm can remove this possibility by further
constraining payments to be either k or l. Indeed, PWYW strategies in the field often require a
minimum price (e.g. Regner and Barria 2009), which is a FP-PWYW hybrid.
2. Probabilistic punishment. Consider the following modification to S’s punishment strategy,
which we call probabilistic punishment with upper thresholdπ , lower thresholdπ , and m
punishment periods: S announces π andπ , with 0 <π <π , such that, if S allows PWYW in
period t and total payment turns out to be tΛ in that period:
(a) If π<Λ t , S switches to FP with posted price p̂ for m periods from t+1 onwards and reverts
to PWYW in period t+m+1;
(b) If π≥Λ t , S continues PWYW in t+1;
(c) If ππ <Λ≤ t , S continues PWYW in t+1 with probability )/()( πππ −−Λ t but implements
punishment as in (a) otherwise (i.e. with probability )/()( πππ −Λ− t ).
14 For a firm S with time discount factor, δS, the time-discounted average of a temporal stream of (expected) incomes
x1, x2, … xt, … is defined as ∑∞
=
−−1
1)1(t
ttSS xδδ .
19
In Appendix C, we show that PWYW-sustaining consumer-side equilibria under probabilistic
punishment, if they exist, generate per period profitπ ; we also present conditions under which
there exist PWYW equilibria with at least as much profit as under perpetual FP.
Probabilistic punishment is a coordination mechanism in the sense that it can be used to
protect an established PWYW equilibrium from crumbling due to some consumers paying less
than what they should. This function is important because, once total payment under PWYW in
a PWYW equilibrium with non-probabilistic punishment falls below threshold, all consumers
may be less inclined to sustain PWYW after the punishment periods are over, thus leading to a
convergence towards the baseline equilibrium. Under probabilistic punishment, even if total
payment falls slightly below the upper threshold, there is still a high probability that PWYW will
continue into the next period. Moreover, as we point out at the end of Appendix C, given that i’s
equilibrium payment is strictly less than an upper bound, as long as the sum of other consumers’
payments under PWYW never falls too far short of that under equilibrium, i still finds following
the PWYW equilibrium to be always preferable to paying zero. This observation that does not
apply to the “step-function” punishment discussed elsewhere in this paper. Thus probabilistic
punishment provides additional robustness for PWYW equilibria over “step-function”
punishment.
3. Consumer communication. In our model, the game that consumers play has the same
flavor as the provision point public goods game. In that literature, van der Kragt et al. (1983)
observe empirically that allowing subjects to communicate before making decisions yields a
much higher success rate of provision relative to when subjects are not allowed to communicate.
We thus surmise that communication will have a similar effect in coordinating consumers to pay
according to a particular PWYW equilibrium. This is by no means a foregone conclusion. In his
20
survey chapter on public goods experiments, Ledyard (1995, p.158) points out that the evidence
on group gains from communication in environments with thresholds (as in our study) is mixed.
Van der Kragt et al. (1983) report that communication increases efficiency and contribution,
while Chamberlin (1978) and Palfrey and Rosenthal (1991) report no discernible effect. Ledyard
concludes that the effect of communication in environments with thresholds “needs more study.”
4. Suggested payments. If, in addition to the threshold and the number of punishment periods,
the firm recommends a suggested payment scheme to consumers which constitutes a PWYW
equilibrium, the suggested scheme may become a “focal point” (Schelling 1960) and thus
function as a coordination mechanism. This is especially likely when the suggested payment
scheme is perceived as a fair and natural choice of equilibrium, such as when every consumer
(regardless of type) has the same per period utility gain with the scheme. The “equi-earnings”
equilibrium that we use in our experiment provides one such numerical example.
To sum up, we have suggested four mechanisms that firms can adopt to enhance consumer
coordination. The functions of payment constraints and probabilistic punishment have been
demonstrated theoretically. However, communication and suggested payments are behavioral
strategies, the effectiveness of which can only be studied empirically. In the next section we
describe an experiment designed to assess whether these elements of coordination are needed for
the sustainability of PWYW15.
15 Although payment constraints and probabilistic punishment should be effective in enhancing coordination based on theoretical reasoning, this does not eliminate the need to verify these predictions experimentally. However, we have elected to experimentally test the effectiveness of the behavioral strategies only, because: (1) they cannot be justified on theoretical grounds alone, (2) they are the simplest to implement, and (3) they maintain the spirit of PWYW to the fullest. Clearly, experimentation with payment constraints and probabilistic punishment are warranted for future studies.
21
3. An Empirical Examination
3.1 Overview of Our Approach
Our model is intended to be prescriptive for the firm. We do not claim that our model describes
the behavior of firms that offer PWYW, nor is it possible to demonstrate that our model is
isomorphic with the various illustrations we provided earlier, to motivate our study. We do,
however, propose that the PWYW strategy can be profitable for firms if the firm interacts
repeatedly with its customer base, and if that customer base displays certain properties that are
specified in our model. Our theoretical results indicate that under certain specific conditions,
self-interested consumers will find a PWYW pricing scheme with pre-specified threshold and
punishment periods attractive, and will sustain it indefinitely to the benefit of both the firm and
the consumers. For our model to be practically valuable, we need to verify that consumers are
likely to behave according to at least one of the PWYW equilibria that are admitted by the model
given the firm’s strategy. In this respect, to be prescriptive for the firm, the PWYW equilibria
must be descriptive of actual consumer behavior.
The goal of the laboratory experiment reported in this section is to assess: (a) whether
consumer communication and suggested payments, the two behavioral coordination mechanisms
proposed in the previous section, can help consumers align their behavior with a particular
PWYW equilibrium and sustain it indefinitely; and (b) as control conditions, whether (i)
theoretically admissible PWYW equilibria are sustainable without coordination mechanisms, and
(ii) whether PWYW is sustainable even when theory does not admit any PWYW equilibria, such
as when the threshold profit is too high and when there is no punishment. To accomplish these
goals, we study the demand side of our model through an experiment in which subjects, who
play the role of consumers, are exposed to simulated firm strategies. Subjects know that the
22
firm’s strategy is pre-determined and is not being played by other subjects. This approach likely
removes any altruistic, non-economic motivation on the part of the subjects to sustain a PWYW
just to allow other subjects playing the role of the firm to earn their “fair share”, had the firm
been played by other subjects. If the proposed PWYW strategy is shown to be a sustainable in
the lab when such non-economic considerations have been removed, the strategy might be even
more successful in the field, where non-economic factors including fairness and reciprocity may
play a role in consumers’ payments (e.g., Kahneman et al. 1986).
We now turn to a discussion of several conceptual factors that influenced the design of our
experiment.
3.2 Design
Our design incorporated several elements. First, across all conditions, we controlled for the
demand profile of the consumer market, and kept the punishment scheme simple for subjects to
understand by simulating a “grim trigger” strategy, according to which the firm would switch to
fixed pricing forever, if total PWYW payment failed to reach threshold in a period. This
corresponds to the case when m tends to infinity in our model. Second, we varied the threshold
π , the other parameter of the PWYW strategies that we examine, across conditions. Third, we
examined whether communication among consumers and suggested payment amounts could be
effective coordination mechanisms. Finally, we considered whether descriptive features of the
experimental stimulus might influence subjects’ ability to generate and sustain a PWYW
equilibrium, by varying the richness of the description of the purchase context embedded in the
experiment.16
16 As we describe shortly, our experimental settings incorporated the context of independent music bands. To eliminate concerns that the “rich” context of indie bands might have primed subjects to employ considerations in their payment behavior that were specific to that particular context, we performed additional empirical work that employed “neutral” stimuli.
23
Model-based factors. We tested three pricing strategies to verify that consumers react to the
threshold and the threat in the predicted manner. Specifically, in Strategy 1, we implemented the
basic model with parameters that satisfy conditions (a) and (b) in Proposition 1, with a threshold
that was set to theoretically admit a PWYW-sustaining consumer-side equilibrium. Strategy 2
was similar to Strategy 1, except that the threshold was set at a high level that theoretically did
not admit PWYW equilibria. In Strategy 3, the threshold was set to zero, which corresponds to a
firm adopting an unconditional PWYW strategy. Of these three pricing strategies, only Strategy
1 theoretically admits PWYW-sustaining consumer-side equilibria.
Note that, if strategic considerations drove subjects to pay non-zero sums under Strategy 1,
then strategic considerations ought to drive subjects to pay zero under Strategies 2 and 3. If, on
the other hand, subjects were motivated by non-strategic considerations such as altruism and
fairness, and therefore paid positive amounts under Strategy 1, then those non-strategic
considerations ought to drive them to pay a positive amount under all three strategies.
Behavioral coordination mechanisms. As discussed above, we manipulated two factors that
may enhance coordination: communication (players were allowed to chat on-line, or not) and
suggested payments (provided, or not provided). Note that, although communication and
suggested payments under Strategy 1 were expected to enhance subjects’ ability to coordinate
and hence sustain PWYW, we expected that, when there was no self-interest based reason to
sustain PWYW (under Strategies 2 and 3), communication and suggested payment should not,
by themselves, be sufficient to generate a sustainable PWYW. That is, communication and
suggested payments can facilitate coordination when PWYW is feasible, not otherwise. If this
expectation turns out to be the case, it strengthens our claim that strategic (rather than non-
economic) drivers are necessary to sustain PWYW under Strategy 1.
24
Context. We manipulated descriptive elements of the experimental context. While the
subject’s task was to purchase online music in one set of conditions, we provided a relatively
sterile or neutral purchase context in another. Subjects exposed to a “rich” context may
potentially be influenced by the context per se rather than by the strategic nature of the situation
(e.g. Chou et al. 2009). Conversely, subjects exposed to a relatively neutral stimulus might focus
more on the strategic elements of the task. This manipulation, therefore, allows for the
identification of a boundary condition for the model, if the sustainability of PWYW is dependent
on the richness of the purchase context.
Figure 2 summarizes our design. Under Strategy 1, two levels of context (rich versus neutral)
were crossed with three levels of coordination, resulting in conditions labeled “No
Chat/Suggestion,” “Chat/No Suggestion,” and “No Chat/No Suggestion”, where “Suggestion”
indicates the provision of a suggested payment scheme 17. Under strategy 2, two levels of context
(rich versus neutral) were combined with the Chat/No Suggestion condition. Six groups of
subjects in each of the above combinations participated under the rich context manipulation and
two groups participated under the neutral context manipulation. In addition, two groups of
subjects participated under Strategy 3, which employed the rich context with the
Chat/Suggestion manipulation.18
17 A “Chat/Suggestion” condition would have made the design a complete factorial design, but this cell was not included because any information from that cell would have been redundant, in light of the information we would obtain from the chat condition. 18 A complete factorial design incorporating all three factors would have been prohibitively complex. Further, Strategy 1 is the option that represents our principal strategy of interest, whereas Strategies 2 and 3 serve as controls. We selected the environment for Strategies 2 and 3 that was most likely to sustain PWYW. Therefore, they likely would provide a strong test for our claim that strategic considerations are necessary to drive payments under Strategy 1, if indeed, subjects reverted to zero payment under Strategies 2 and 3.
25
---------------------------------------
Insert Figure 2 about here
---------------------------------------
3.3 Experimental Procedures
Subjects were divided into groups of eight, and every subject interacted with the same group of
seven other subjects throughout the session. During the experiment, all decisions were made via
networked computers using the z-Tree software (Fischbacher 2007). Two hundred seventy-two
undergraduate subjects from a university in Hong Kong participated in the experiment. All the
subjects volunteered to participate in the study, which was billed as a decision-making
experiment with payoff contingent on performance.
Each session consisted of a practice game followed by 20 games. Each game consisted of an
indefinite number of rounds such that, after every round, there was a 90% probability that the
game would continue to the next round and a 10% probability that the game would end
immediately. This procedure thus created an effective per period time discount factor of δ = .9
for the subjects. This is a common procedure employed in laboratory experiments designed to
mimic infinitely repeated games with time discounting (Zwick et al. 1992). In every condition,
after all the games were concluded in a session, 5 games were chosen at random from the 20
games played and each subject was paid his/her earnings from all the rounds in those games after
converting tokens (the experimental currency) to Hong Kong dollars (US$1 = HK$7.8).19 In
addition, every subject was paid a show-up fee of HK$40. Average subject payment across all
conditions, including the show-up fee, was HK$171.7.
19 The conversion rate is 1 token = HK$0.1 for the three conditions with threshold = 400 tokens, and 1 token = HK$0.4 for the condition with threshold = 680 tokens. These rates make the expected payoff of the Pareto optimal equilibria in all main conditions approximately the same. The conversion rate is 1 token = HK$0.1 for the no threshold condition.
26
For expository convenience, we first describe the experimental context.
Context. Recall that context was manipulated by providing subjects rich information or
neutral information. The actual instructions used in both contexts are included in Appendix D.
Rich context. Subjects were told that two (fictitious) bands, “Playa” and “Quello”, each
uploaded a new song to their site during every round of the game, to allow people to listen to it
online. During each round, a player could either listen to a Playa song or to a Quello song, or to
none. At the beginning of each game, each player was assigned to one of two types: “Fan of
Playa” or “Casual Listener”. These categories corresponded to the “fan” and “casual” segments
in our model, with Playa representing the firm S, and Quello representing the outside option.
The assignment of types was such that, of the eight players in each group, two were Fans of
Playa and six were Casual Listeners, yielding q = 1/4. Every player’s role stayed the same in all
rounds of the same game, but was re-assigned randomly from game to game, with the constraint
that each player had to be a Fan of Playa in 5 games and a Casual Listener in 15 games. The
valuations of the different types of players towards the different bands’ music follow the model
with 201=Fu tokens and 49=Cu tokens. The price of the outside option, i.e. Quello, was
480 =p tokens. Subjects were told that, in the first round of every game, Playa allowed every
listener to pay as s/he wished, i.e., Playa’s first-round strategy was always PWYW. Moreover, if,
in a round with PWYW, the total payment that Playa received from its listener(s) reached a
threshold number of tokens (400 under Strategy 1 and 680 under Strategy 2), Playa would
continue to allow every listener to pay as s/he wished in the next round (if the game continued
into the next round), but if Playa received less than the threshold amount in a round, it would
change its pricing scheme and would charge every listener a fixed price of 200 tokens per round
in all future rounds (if any) of the same game. In every round of the game, a player needed to
27
choose one from the following set of options: listen to a song by Playa, listen to a song by Quello,
or listen to neither. If Playa employed FP in that round and if Quello or Playa were chosen, the
player earned a net payoff that was equal to her valuation of the song of her choice minus its
(fixed) price. If Playa allowed PWYW in that round and the player decided to listen to a Playa
song, she entered the price she decided to pay for the song and she earned a net payoff that was
equal to her valuation of the Playa song minus the price she decided to pay for it (see Appendix
E for sample screenshots of the decision screens). After a round in which Playa allowed PWYW,
all players were informed about the decisions of all other players, though players were not
identifiable and so the decision history of individual players could not be traced by others.
Players were also informed about whether total payment reached the threshold and whether
Playa would continue to allow PWYW in the next round. A randomization process then took
place to determine if the game proceeded to the next round or was terminated.
Neutral context. The instructions were structurally similar to those provided in the “rich”
condition, except that no specific context was provided. For example, subjects were given
options named “S”, “N”, and “R” to choose from, instead of “Playa”, “Neither”, and “Quello”.
Their experimental task was described plainly as a choice between three options rather than a
choice between listening to a song from Playa, Quello, or neither. At the beginning of each
game, subjects were assigned to be of either “Type Y” or “Type Z”, instead of “Fan of Playa”
and “Casual Listener”. The threshold was described in neutral terms: PWYW continued in the
next round with option S iff (1) the payment scheme for S was PWYW in the current round and
(2) total payment to S reached a pre-announced threshold in the current round. However, context
description apart, the experimental treatment was identical between groups playing within every
condition.
28
Strategies. The three strategies were operationalized as follows.
Strategy 1. In this condition the game’s parameters satisfied conditions (a) and (b) in
Proposition 1 and the threshold was set to allow a theoretically sustainable PWYW. The
parameters were: N = 8, δ = .9, q = 1/4, 201=Fu tokens, 49=Cu tokens, and 480 =p tokens.
Since 25.50=Fqu tokens 0p> , and { } 1545.1]/)1[(1 0 >=−+ Fqupqδ , conditions (a) and (b) in
Proposition 1 were satisfied. Subjects were told that, in the first round of every game, they could
pay as much as they wished for the offering from firm S. Moreover, if, in a round with PWYW,
the total payment to S reached 400 tokens (π =400 tokens), PWYW would continue in the next
round (if the game continued into the next round), but if the total payment turned out to be less
than 400 tokens in a round, S would change its pricing scheme and would charge a fixed fee of
200 tokens per round in all future rounds (if any) of the same game ( p̂ =200 tokens). From
Proposition 1, we know that PWYW-sustaining consumer-side equilibria with theoretical
threshold profit (after accounting for the total number of consumers) is between 402 (= Fqu8 )
and 621 (= ])1([8 0pqquF −+δ ) tokens; however, for practical reasons, we used a threshold profit
of 400 tokens.20
Strategy 2. The purpose of this condition was to eliminate the possibility that non-strategic
considerations might drive a sustainable PWYW outcome. Such an outcome might occur if
subjects adhere to a payment scheme that is in fact not incentive compatible in terms of their
own monetary payoffs, but which allows them to derive extra utility due to the sheer ability to
20 According to theory, the per period optimal price for S should indeed be 201 tokens, and thus, once S turns to FP, it should charge 201 tokens rather than 200 tokens, and earn a per period profit of 402 tokens. However, in the experiment, to make sure that there was no tie between any two alternatives under FP, we used 200 tokens as S’s FP (which means that its per period profit under FP was 400 tokens) so that Fans were expected to strictly prefer S to no purchase (and the outside option) when S switched to FP. This does not alter our conclusions based on Proposition 1. What it amounts to is simply changing the effective uF to 200 tokens for ease of calculation.
29
sustain PWYW through cooperation. In such an event, a high threshold would also yield
PWYW outcomes.
Recall that per period profit under any PWYW equilibria must be no more than 621 tokens, if
we apply Proposition 1 to our parameters. In this condition, we raised the threshold profit
requirement to 680 tokens (π =680 tokens). Thus PWYW should be unsustainable unless non-
strategic considerations play a role in consumers’ decisions.
Strategy 3. This strategy was similar to Strategies 1 and 2 except that the threshold was set to
zero (π =0 token). That is, S implemented PWYW unconditionally whatever payments (if any)
it received from the players. Although the threshold was effectively set at zero, subjects were
told that the focal firm’s management hoped to receive at least 400 tokens every round from their
customers. This experimental condition is important because our proposed PWYW strategies are
crucially dependent on consumers’ reaction to the firm’s use of the threat to switch to fixed
pricing in all future periods if the threshold is not met. Since there is no such threat in this
condition, if we find that consumers pay positive prices under Strategies 1 and 3, our premise
that subjects reacted to the firm’s threat of employing a fixed price in the future would be
questionable. If, on the other hand, PWYW was sustainable under Strategy 1 but consumers
reverted to paying zero under Strategy 3, we can infer that consumers did respond to the firm’s
threat to switch to fixed price in the future (i.e., strategy 1).
Coordination mechanisms. The two coordination mechanisms were operationalized as
follows.
Communication. In the no-communication condition (no chat condition) subjects made their
decisions without communicating with each other. In the communication condition (chat
30
condition) subjects in the same group were allowed to engage in an online chat in a “chat forum”
before the first period of every game for a limited duration.21
Suggested payments. In conditions with suggested payments, all subjects were provided a
suggested payment scheme for sustaining PWYW. The suggested scheme appeared on every
decision screen and was also mentioned in the instructions. For example, under the rich frame
manipulation, subjects read the following instructions:
Suggested payments under the “pay as you wish” scheme Although Playa allows players to pay as much as they wish for listening to their song in a round when the “pay as you wish” scheme is implemented, they nevertheless provide suggested payment amounts. In particular, they suggest that Fans of Playa pay 164 tokens and hence earn (201-164=) 37 tokens in such a round. Causal Listeners are suggested to pay 12 tokens and hence earn (49-12=) 37 tokens in such a round. If all players pay their suggested amount in such a round, the total payment to Playa will be (2x164+6x12=) 400, exactly the amount needed to keep the “pay as you wish” scheme going to the next round (if there is a next round).
3.4 Analysis and Results
All analyses and results reported here exclude data from the practice game, since subject
behavior in the practice game was not incentivized. Our dependent variables of interest are the
mean payment for the focal firm’s product in a PWYW round, and the number of rounds with
sustained PWYW (a round with sustained PWYW is one in which PWYW is allowed and total
payment to the focal firm reaches the threshold). We first describe our statistical analyses and
then provide some insights that emerge from an examination of the chat log.
Statistical analyses. We first test for the effect of context. We conducted an ANOVA
employing a 2 (context: rich or neutral) x 3 (communication and suggested payment
21 We ran a pilot session with unlimited chat allowed throughout the experiment; the results were similar to the chat condition reported here, but it took subjects almost three hours to finish the session. Therefore, for practical reasons, we chose a limited duration structure for our main experiment. The allowed duration for chat before each game was: (1) Practice Game to Game 6: 3 minutes per game; (2) Game 7 to Game 13: 2 minutes per game; (3) Game 14 to Game 20: 1 minute per game.
31
combinations: No Chat/No Suggestion; Chat/No Suggestion; No Chat/Suggestion) design under
Strategy 1, on the two dependent variables. We find that there is no significant main effect of
context (F(1,18) = 1.02, p > .3 and F(1,18) = 1.82, p > .1), nor is there a significant interaction
(F(2,18) = 0.79, p > .4 and F(2,18) = 1.45, p > .2) under Strategy 1 for both the overall mean
payment and number of sustainable PWYW rounds, respectively. We also analyzed a simple
effect of context under Strategy 2 and found no significant main effect (F(1,6) = 3.28, p > .1 and
F(1,6) = 1.27, p > .3), for both variables. Thus we conclude that context has no effect on the
dependent variables, and therefore, for the remaining analyses, the data from the two contexts is
combined.
Table 2 presents the main results (aggregated over context). A preliminary inspection of
Table 2 reveals that mean payments in a PWYW round (average over consumer type: 49.40
tokens) and sustainability (167.50 rounds) is much higher under Strategy 1 Chat/No Suggestion
condition than under any other condition, except, of course, for sustainability under Strategy 3.
In that condition, because of the zero threshold, any payment (including zero) yields a
sustainable outcome.
------------------------------------------
Insert Table 2 about here
------------------------------------------
The importance of coordination is confirmed by further statistical analysis on the mean
payment in PWYW rounds and the number of rounds with sustained PWYW. A 1-factor
(coordination type) MANOVA on both dependent variables reveals significant main effects
across the three coordination conditions under Strategy 1 (Wilks’ λ = .087, F(4,40) = 23.81, p
<.0001). This is confirmed by a univariate ANOVA for both variables (F(2,21)=5.19, p = .015
32
for mean payments in PWYW rounds, and F(2,21)=91.01, p < .0001 for sustainability,
respectively). Pair-wise comparisons using MANOVA confirm that both variables are indeed
significantly higher in the Chat/No Suggestion condition than in any of the other two conditions
(Chat/No Suggestion vs. No Chat/No Suggestion: Wilks’ λ = .027, F(2,13) = 234.24, p <.0001;
Chat/No Suggestion Vs No Chat/Suggestion: Wilks’ λ = .12, F(2,13) = 45.99, p <.0001). These
results are confirmed by corresponding univariate pair-wise comparisons with each dependent
variable (p< .05 in all comparisons). A pair-wise comparison between the Chat/No Suggestion
conditions under Strategies 1 and 2 also shows a significant effect (Wilks’ λ = .028, F(2,13) =
221.70, p <.0001) that is confirmed by univariate pair-wise comparisons of both dependent
variables (p<.01 in all comparisons).
Further pair-wise comparisons using MANOVA reveal no other significant effects between
conditions in Strategies 1 and 2 at p < .1. But univariate comparisons involving only the number
of rounds with sustained PWYW show that No Chat/Suggestion in Strategy 1 yielded higher
sustainability of PWYW than the No Chat/No Suggestion condition in Strategy 1 and the
Chat/No Suggestion condition in Strategy 2, with marginal significance (F(1,14)=3.35, p = .089
and F(1,14)=3.49, p = .083 respectively). The first result implies that a suggested payment has
some positive impact on PWYW sustainability compared with no suggestion or when the
threshold is too high (even if consumers can communicate with each other in the latter case). No
other pair-wise comparisons of either of the two major dependent variables among conditions in
Strategies 1 and 2 yield effects at the p < .1 level.
It is also clear from Table 2 that unconditional PWYW (Strategy 3) yielded much lower
payments than conditional PWYW that allows for theoretically sustainable PWYW (Strategy 1).
A pair-wise comparison between the conditions in Strategy 1 and Strategy 3 shows that mean
33
payments in PWYW rounds are significantly higher in any of the former than in the latter (No
Chat/No Suggestion: F(1,8)= 8.71, p = .018, Chat/No Suggestion: F(1,8)= 4021.05, p < .0001,
No Chat/Suggestion: F(1,8)= 19.27, p = .0023). But there is no significant difference in mean
payments between the conditions in Strategies 2 and 3 (F(1,8)= 1.47, p > .2).
Figure 3 presents the mean number of rounds with sustained PWYW classified by the game’s
length (in rounds). For purposes of reference, under perfect sustainability, a line representing
y=x is also provided. Under Strategy 1 Chat/No Suggestion condition, the plot is much closer to
the perfect sustainability line than the plots associated with the other conditions, providing a
visual illustration of the importance of both the threat of a fixed price and the availability of chat
in sustaining PWYW.22
------------------------------------------
Insert Figure 3 about here
------------------------------------------
Based on these analyses, we conclude that the experimental data demonstrates the feasibility
of the proposed PWYW policy. That is, within the context of our experiment, a PWYW strategy
with a pre-announced profit threshold induces highly sustainable PWYW iff: (1) there exists an
individually incentive compatible payment scheme that sustains PWYW (i.e. PWYW-sustaining
consumer-side equilibria are theoretically sustainable given the strategy), and (2) subjects can
communicate with each other to coordinate among themselves to sustain PWYW.
How does communication function as a coordination mechanism? Based on van der Kragt et
al. (1983), we suggest that, first of all, communication allows subjects to align their strategies
along one payment scheme; secondly, communication helps subjects to make a commitment to
each other on adhering to an agreed upon payment scheme. To further examine these proposed 22 Strategy 3 is not included in the figure since sustainability is not a valid indicator.
34
functions and the nature of communication that occurred during the chat periods, we next turn to
an analysis of the chat log.
Thick Description. The following qualitative insights emerge from an examination of the
raw data and the chat log:
(1) Under fixed pricing, subjects paid according to predictions (i.e. Fans of Playa chose Playa
and Casual Listeners chose Quello) in at least 96% of the observations in any condition;
(2) In Strategy 1 Chat/No Suggestion condition, six groups played according to the equi-earnings
equilibrium. The remaining two groups played according to almost equi-earnings equilibria,
with the per round payments of Fans of Playa/Casual Listeners being 170 tokens/10 tokens
for one group and 162 tokens/13 tokens (which make up a total payment of 402 tokens) for
another group;
(3) It appears that many subjects intuited from early on that they should cooperate to sustain
PWYW under Strategy 1 (π =400). This is most directly reflected in the chat log in the Chat
condition in which there was hardly any challenge to the notion that subjects should
cooperate. Subjects were instead occupied from early on with arriving at a commonly agreed
upon payment scheme through chat. Here are some suggestive quotes:
a. “ …[if] anyone cheat (sic) all of us get the least ... including the cheater,”
b. “ If we know when [the game ends] we can simply pay zero [in the last round] ...
but [since we do not, it is] not worth taking the risk,”
c. “… please think of the benefit of the whole team,”
d. “… if you break our relationship ... you will earn less,”
e. “PLEASE DON'T TRY TO CHEAT!!!!!!!!!!!!!!!!!”
f. “…the more the number of round[s] the more we gain.”
35
(4) Although our experiment was not designed to explore how chat enhances the sustainability of
PWYW, nevertheless, a causal inspection of the chat log in the Chat condition of Strategy 1
seems to support the assertion provided by van der Kragt et al. (1983) that chat enhances
cooperation in two major ways:
(a) Subjects can align their strategies by working out a PWYW-sustaining payment scheme
that just sums up to the threshold and which everyone agrees upon. Such a payment
scheme is essentially what van der Kragt et al. call a “minimally contributing set”, as no
subject has an incentive to deviate by paying less than specified in the scheme. In the
experiment, this was usually achieved by one player suggesting the payment scheme and
then discussing with/explaining to other players why they should follow it;
(b) Subjects can make commitments to each other that they will adhere to the payment
scheme that is agreed upon. In other words, chat allows for the establishment of an
obligation or a “social contract”, even though chat is essentially “cheap talk” and, if
individual deviations in a game occur, it was impossible to identify renegades who might
have violated their commitment to pay their share. It was typical that, following the
suggestion and discussion of a particular payment scheme, every player would send out a
single-line message as a confirmation to others of his/her agreement with the scheme,
before the chat session was concluded for the upcoming game. Also, since the No
Chat/Suggestion condition under Strategy 1 did not lead to highly sustainable PWYW, a
“social contract” is apparently essential for PWYW sustainability in the chat condition;23
(5) In the No Chat/No Suggestion condition under Strategy 1, subjects appeared to have serious
coordination problems. Attempts to sustain PWYW by individual subjects were very often
23 Since no “social contract” can be established if there is no agreed upon payment scheme to start with, an agreed upon payment scheme is therefore necessary for the manifestation of a social contract.
36
undermined by other subjects’ low or zero payments. Even in the No Chat/Suggestion
condition under Strategy 1, initial enthusiasm to sustain PWYW could have been dampened
because certain subjects tried to take a little advantage by paying slightly less than what was
suggested for their role, despite the very clear realization that everyone’s payment was
critical to sustaining PWYW. However, attempts to establish sustained PWYW were
observed throughout the session in both No Chat conditions of Strategy 1;
(6) Under Strategy 2 (Chat/No Suggestion), the data show that subjects in general quickly
arrived at the understanding that it was best for them not to sustain PWYW and pay zero
tokens for Playa’s song in every game’s first round; and
(7) Subjects under Strategy 3 always took advantage of Playa and paid close to zero for their
song. The mean (standard deviation) individual payment was 1.85 (2.14) tokens (average
over both types). Although subjects were told that Playa’s management hoped to receive at
least 400 tokens every round, the mean total payment was always far lower than that.
Subjects paid zero tokens to Playa in 69.97% of the observations. There are only 46
observations (i.e. 1.51% of all observations under Strategy 3) of a subject paying according
to the suggested equi-earnings scheme. We conclude that, when there was no threat to
switch to fixed price if total payment did not reach the threshold, subjects almost never
collectively paid under PWYW to reach 400 tokens, despite Playa’s plea for a payment of
400 tokens every round. Seemingly, non-economic factors such as altruism and fairness
were not the main considerations among our subjects.
4. Concluding Remarks
4. 1 Summary
37
In this paper, we first develop a simple economic model to investigate whether and how a firm
can continuously earn a profit by allowing self-interested consumers to pay whatever they want
for its products. Our model incorporates an infinitely repeated pricing game with (effectively) a
two-segment market, in which the firm could use a switch from “pay what you want” to fixed
pricing over a pre-specified number of “punishment periods” as a threat to incentivize consumers
to pay a sufficiently high price even under PWYW. Our model predicts that a market of
consumers admits a PWYW sustaining consumer-side equilibrium when the consumers are very
forward looking and a portion of them are “priced out” under a fixed price only regime, because
the firm earns a slightly higher profit by setting a high price relative to when it prices sufficiently
low to capture all consumers.
We then offer four coordination mechanisms that are aimed to avoid coordination failures in
sustaining PWYW. Two of these are justifiable on theoretical grounds while the other two,
namely that consumers are allowed to communicate before making payment decisions and that
consumers are provided suggested payments, are behavioral strategies inspired by previous
literature and require empirical verification. We therefore experimented on the latter two
strategies, and concluded that, while suggested payments did not lead to a sustainable PWYW
equilibrium, consumer communication proved to be an effective coordination mechanism.
Overall, within our experimental conditions, a PWYW strategy induced highly sustainable
PWYW among consumers iff: (1) the strategy theoretically admits PWYW-sustaining consumer-
side equilibria, and (2) subjects were allowed to communicate with each other.
4.2 Implications
Theoretical issues. According to our model, it is sufficient that the firm sets a PWYW
“target” or “threshold” that it estimates is at least as much as the amount it would earn under
38
other pricing schemes, announce this threshold to consumers, and alerts them to actions that it
will take in the future that would adversely affect consumers if the threshold was not reached.
This is a purely self-interest based argument. However, in light of the extant literature that
suggests a role for non-economic factors, we wished to assess whether PWYW can be a
profitable strategy even when consumers are not motivated by altruism or their innate sense of
fair play. Consequently, in our experiment, we attempted to eliminate the potential impact of
non-economic factors by having subjects play against a simulated firm rather than another
subject.
Whether non-strategic factors are sufficient to behaviorally sustain a common, unconditional,
PWYW scheme (such as under Strategy 3) is an open question. Our speculation, based on our
results (recall that Strategy 3 failed to sustain PWYW outcomes) is that non-economic factors
such as altruism, a fairness motive, and the like, are insufficient to sustain PWYW as a long term
pricing policy.
However, if the conditions specified in our model are fulfilled, the likelihood of PWYW
being a sustainable profit-generating pricing strategy is likely to be enhanced when consumers
are altruistic as well. That is, non-economic factors that foster the norm of reciprocity and fair
play will likely improve the firm’s profits under a PWYW pricing environment.
Managerial issues. Our results suggest that for PWYW to succeed, the firm must clearly
announce its intention to switch from PWYW to a fixed price over a pre-specified duration24
should a pre-specified revenue threshold not be achieved, and this explicit threat must be
perceived to be credible by consumers. Furthermore, the PWYW strategy does not have to be a
loss leader strategy that needs to be subsidized by derivative, secondary revenue. In markets in
24 It suffices if this pre-specified duration is understood by the consumers to be the minimum (as opposed to precise) duration of fixed price punishment. It is straightforward to deduce that a PWYW equilibrium satisfying Proposition 1 with threshold π and m punishment periods remains admissible when m is longer, controlling for π.
39
which there is a segment of “die-hard” consumers, who can potentially influence casual
consumers, PWYW is a feasible pricing strategy, so long as there is repeated interaction between
the firm and consumers, and our other modeling assumptions are not violated.
These theoretical conditions are likely to be observed in the case of independent music bands,
particularly when their fans communicate with each other and with casual listeners, since such
communication can serve as an effective coordination mechanism. A small group of lovers of
non-mainstream music likely constitute the band’s core customers, who form a closely knit
social network with frequent communication through blogs, social network sites, tweets, and the
like. Further, fans probably have the ability to influence a “casual” segment (comprising friends
and relatives) to patronize the band and pay sufficiently high prices, so that fans may avoid
paying a high fixed price in the future.25 Similarly, a neighborhood restaurant serving a local
community might meet the criteria as well. The restaurant could build a brand reputation for
being sensitive to social welfare-issues, and thus attract a “fan base” who then influence their
social network to patronize the restaurant and pay sufficiently high prices to avoid the prospect
of a higher fixed price. In addition, the firms in these examples can restrict the range of
acceptable payments, a regime that is relatively easy to implement especially in online
transactions, and probabilistic punishment, to help consumers sustain PWYW. Note that, in
practice, in a probabilistic punishment world, consumers need only infer that the firm would
continue PWYW with some likelihood even if total payment does not reach threshold, but this
likelihood will decrease as the gap between the total payment and the threshold increases.
4.3 Limitations and Future Research
25 For example, the website Kroogi.com, which is a content community where musician, artists, writers and photographers post copyrighted materials, allows consumers to self select themselves to ever expanding “social circles” of fans and to download the materials based on the traditional PWYW model. Online chat forums for various circles of fans are provided by the site developer.
40
Although the roles of constraints on payment amounts and probabilistic punishment as
coordination mechanisms are theoretically justifiable, it would be valuable to verify them
empirically. Meanwhile, our experiment employed only eight consumers in each market.
Laboratory conditions do not allow for the conduct of experiments with group sizes that
approximate naturally occurring markets, so our results may immediately apply only to relatively
niche markets with a small number of potential consumers. Clearly, it would be useful to
conduct field experiments to see if the insights we gain from our experiment generalize to larger
markets.26 More crucially, our idea of increasing consumers’ incentive to pay (in fact,
incentivizing even the most “selfish” consumers to pay) by communicating a credible threat has
not been tested empirically in the field.
In face-to-face interactions, consumers might develop “reputations” which ought to limit
free-riding and thus reduce coordination problems. Consumers may also derive utility from
PWYW pricing schemes and may wish to encourage the firm to continue using this mechanism,
and therefore may choose to pay sufficiently high prices to sustain a PWYW outcome. For
example, 27% of the consumers (out of 12,643) who responded to a survey after paying what
they wished while downloading the “World of Goo” game selected the option “I like the pay-
what-you-want model and wanted to support it” in answer to the question “Why did you choose
that amount?”27. Future modeling work might take these factors into account to add richness to
the theoretical argument and practical applicability of PWYW strategies.
26 The fan base need not be very large for profitable application of our proposed pricing scheme. For example, musician Matthew Ebel said that he makes 26.3% of his net income from just 40 hard-core fans (http://www.musicthinktank.com/blog/in-defense-of-1000-true-fans-part-ii-matthew-ebel.html) which relates to Kevin Kelly's theory that to be a success as a content creator, you just need 1,000 "true fans." (http://www.kk.org/thetechnium/archives/2008/03/1000_true_fans.php). 27 We thank Ron Carmel and Kyle Gabler, the 2D Boy team, for sharing the survey data with us.
41
In our model, the firm’s threat is the institution of an unpalatable fixed price regime, should a
threshold level of profit not be achieved. An alternative and viable threat in some instances (e.g.,
non-governmental organizations) is the prospect of the firm closing its shutters. This scenario
can be formulated as a game similar to our model. Exiting from the market can be used as a
threat to induce consumer-side equilibria that sustain PWYW. A simple adaptation to our
standard model would be: (1) maintain the setup of a potential market of “fans” and “casuals”
with different valuations Fu and Cu respectively, towards the organization’s offering ( CF uu > );
(2) however, no consumer in the potential market has any outside option, the organization’s
offering being considered unique (competition between organizations can also be modeled); (3)
the threshold is then taken as the per period fixed cost of operation, T; (4) for expository
purposes, simply assume that once the organization exits from the market, it exits forever, so that
all consumers in the potential market earn zero utility in all subsequent periods. Then, a
calculation similar to that employed to prove Proposition 1 shows that, if Tuquq CF ≥−+ ])1([δ ,
a PWYW strategy with threshold π such that Tuquq CF ≥≥−+ πδ ])1([ admits PWYW-
sustaining consumer-side equilibria while generating sufficient revenue for the organization to
survive.28
28 A related tactic was used in 2000 by Stephen King when he started publishing The Plant as an electronic serial. On his website he promised to unconditionally publish the first two installments (by July 24 and August 21, 2000). He asked that readers pay at least $1 for each download under an honor-system payment model. That is, payment was not enforced and readers could have chosen to pay nothing and have still downloaded the first two installments. King further promised that if pay-through equaled or exceeded 75%, Installment Three would go up in September. As long as pay-through equaled or exceeded 75%, King promised to carry The Plant through to its conclusion. However, if pay-through fell below the set threshold, King committed to pull the plug on the story. In his words "If you pay, the story rolls. If you don't, the story folds." The pay-through rates for installments 1 and 2 were 75% and 70%, respectively. For the third installment the rate rose back to 75% but for the fourth installment the pay-through rate dropped to 46% and King pulled the plug on the book. The book has yet to be completed. Other authors have used similar online publishing methods with varying degrees of success (e.g., Lawrence Watt-Evans’ The Spriggan Mirror and Diane Duane's The Big Meow.) Note, however, that in this model the value of an installment may be higher for those who have already invested in earlier ones.
42
Finally, model extensions could incorporate competition, and each of the competing firms
can decide between PWYW and FP for its product in every period of an infinitely repeated
pricing game. The model might also consider a highly stochastic market environment in which
consumer tastes and outside options change from period to period. In these scenarios, we still
expect that equilibria with profitable PWYW could exist over some parameter ranges, as long as
sufficient numbers of repeated interaction occur and FP can be used as a credible threat.
However, coordination issues remain, and experiments on these extended models could yield
insights on how PWYW could be sustained in such highly complex business environments.
43
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Table 1
Consumer valuations in the standard model
S’s product The outside option
Fan Fu 0
Casual Cu Cu
Table 2
The mean payment in a PWYW round and mean number of rounds with sustained PWYW
Mean payment in a PWYW round
(s.d.)
Mean no. of rounds (s.d.)
Strategy Fans Casuals
played with
sustained PWYW
1
No Chat No Suggestion 75.96
(39.84) 18.41
(14.50) 191.50
(4.87) 10.50 (9.96)
Chat No Suggestion 160.76
(4.96) 12.28 (1.01)
193.00 (5.15)
167.50 (18.24)
No Chat Suggestion 117.61
(41.01) 11.36 (4.31)
191.88 (4.67)
35.88 (37.95)
2 Chat
No Suggestion 48.06 (57.01)
14.25 (12.93)
192.75 (4.50)
9.25 (13.65)
3 Chat
Suggestion 3.87 (3.84)
1.17 (1.58)
190.00 (0.00)
190.00 (0.00)
48
Figure 1
The PWYW pricing game: sequence of decisions in each period.
FP: price p PWYW
Buy from Sat price p
Make No purchase
Buy outside option
Buy from Sat price of
choice
Make No purchase
Buyoutside option
Firm SAnnounces pricing decision
Each consumer decides whether to Each consumer decides whether to
49
Figure 2
The Experimental Design
The common parameters in all conditions are δ = .9, q = 1/4, 201=Fu , 49=Cu , 480 =p and p̂ =200 (except in Strategy 3).
The Pricing Strategies: Strategy 1 In the first round of every game, subjects can pay as much as they wish for firm
S’s offering. Moreover, if, in a round with PWYW, the total payment to S reaches 400 tokens (π =400), PWYW will continue to the next round (if the game continues into the next round), but if the total payment is less than 400 tokens in a round, S would change its pricing scheme and would charge a fixed fee of 200 tokens per round in all future rounds (if any) of the same game ( p̂ =200).
Strategy 2 Similar to Strategy 1 except that the threshold is raised to 680 (π =680). Strategy 3 S implements PWYW unconditionally in every round.
Strategic Factors Non-Strategic Factors
Pricing Strategy Communication and Suggested Payments Context
Strategy 3
Strategy 1
Strategy 2
Chat / Suggestion
No Chat / No Suggestion
Chat / No Suggestion
No Chat / Suggestion
Chat / No Suggestion
Rich
Rich
No
Rich
Neutral
50
Figure 3
Mean number of rounds with sustained PWYW in games with different total number of rounds
0123456789
101112131415
4-5 6 7 8 9 10 11 12 13 14 15
Total no. of rounds
Mea
n no
. of r
ound
s w
ith
sust
aine
d PW
YW
Strategy 1, No Chat / No Suggestion Strategy 1, Chat / No Suggestion
Strategy 1, No Chat / Suggestion Strategy 2, Chat / No Suggestion
Perfectly sustained PWYW
APPENDICES (to be published online as Electronic Companion)
APPENDIX A: Proof of Proposition 1
The aim of the proposition is to outline conditions under which a PWYW strategy with
threshold 0ππ ≥ and m punishment periods admits consumer-side equilibria that sustain PWYW.
So, consider a PWYW strategy with threshold ∆+= 0ππ , where 0≥∆ . Consider payment
schemes according to which consumer i pays the same ip for S’s product in every period with
PWYW. Any one such payment scheme can be represented by the set }{ ip . We shall examine
the necessary and sufficient conditions for the existence of an incentive compatible }{ ip that
sustains PWYW given the threshold ∆+= 0ππ .
We start with the necessary conditions, which will make up the “only if” part of Proposition
1. Suppose an incentive compatible and PWYW-sustaining }{ ip exists. Then, as detailed in
footnote 12 in the main text, the sum of the ip s must add up to exactly the threshold ∆+0π :
(1) ∆+=∑ 0πi
ip .
Moreover, every consumer must find that paying ip every period for S’s products is at least
as good as paying zero for S’s product in a period and then buying S’s products or the outside
option for m periods afterwards under FP. That is:
(2a) If i is a fan, δ
δδδ
δ−
−−+≥−⋅
−− +
1)ˆ)(1()(
11 1 puupu F
m
FiF
m
, or pp m
m
i ˆ1
)1(1+−
−≤
δδδ ;
(2b) If i is a casual, δ
δδδδ
δ−−−
+≥−−
⋅−
− +
1}),ˆmin{)(1(
111 0
1 ppuu
pu Cm
CiC
m
, or
},ˆmin{1
)1(01 ppp m
m
i +−−
≤δδδ ,
where p̂ is the one-period optimal selling price for S, which is also the price it will post under
perpetual FP with constant price. Note that, strictly speaking, we should also write out the
constraints 0≥∀ ipi at this point, but the above conditions certainly satisfy these constraints,
and the following deductions are unaffected as well.
Now, if there exists a payment scheme that satisfies conditions (1), (2a), and (2b), we must
have:
∆+≥⋅−+⋅⋅−−
+ 001 }],ˆmin{)1(ˆ[1
)1( πδδδ ppqpqNm
m
.
It is easy to see that, when 0pquF < , so that Npp /ˆ 00 π== , the above inequality is impossible
to fulfill, given 1<δ and 0≥∆ . Therefore we must have 0pquF ≥ so that Fup =ˆ and
FNqu=0π , and the above condition becomes:
∆+≥−+⋅−−
+ FFm
m
NqupqquN ])1([1
)1(01δ
δδ , or
Nqupqqu FFm
m
/])1([1
)1(01 ∆≥−−+⋅
−−
+δδδ .
Thus, if 0≥∆ , we must have:
0pquF ≥ , and
0])1([1
)1(01 ≥−−+⋅
−−
+ FFm
m
qupqquδδδ , i.e. 1
)1(1
1)1( 0
1 ≥
−+⋅
−−
+F
m
m
qupq
δδδ .
These make up the necessary conditions pertaining to Proposition 1. The upper bounds for
per period payment, i.e., (i) and (ii) in Proposition 1, are simply the results in (2a) and (2b) in
conjunction with the condition 0pquF ≥ . Constraint (iii) in Proposition 1 is simply the
condition (1) stated earlier in this proof.
For the “if” part of Proposition 1, suppose conditions (a) and (b) are fulfilled. Then, by (b), it
is possible to choose π such that FFmm quNpqqu ≥≥−+⋅−− + /])1([)]1/()1([ 0
1 πδδδ . With
such a π , consider the following payment scheme:
Fmm
i up )]1/()1([ 1+−−= δδδε if i is a fan, and
01 )]1/()1([ pp mm
i+−−= δδδε if i is a casual, where
])1()][1/()1([ 01 pqquN F
mm −+−−= +δδδ
πε .
Then (2a) and (2b) are fulfilled if 0pquF ≥ , which is condition (a), and 01 >≥ ε , which is
immediately satisfied by our choice of π . Therefore, if (a) and (b) of Proposition 1 are fulfilled,
then there exists π satisfying FFmm quNpqqu ≥≥−+−− + /])1()][1/()1([ 0
1 πδδδ , and, for
every such π , a PWYW strategy with threshold π admits a set of consumer-side equilibria
(with at least one member as in the one example we just discussed) according to which S always
allows PWYW and makes profit π every period. The payment scheme associated with the
discussed example, moreover, certainly satisfies the constraints (i) to (iii) in Proposition 1.
As an end note to this section, we establish that the conditions (a) and (b) in Proposition 1 are
in fact necessary for a PWYW strategy with m punishment periods to admit any pure strategy
consumer-side equilibria that sustain PWYW and generate at least as much per period profit as
0π . In effect, we need to consider cases when consumers could pay temporally varying prices
under PWYW. Thus far we have only considered equilibria in which each consumer pays the
same amount under PWYW in every period (although payments across consumers can be
different; in fact, since the upper bound for a casual’s payment, 01 )]1/()1([ pmm +−− δδδ , is
lower than Fqu given 0pquF ≥ , there must be fans who pay more than any casual under
equilibrium to assure that the total payment is not lower than Fqu ).
Based on any arbitrary reference period, index the periods by the variable t. Consider a
generalized payment scheme under PWYW with which consumer i pays itp for S’s product in
period t. Then conditions (2a and b) need to be modified as follows:
(2a) If i is a fan, δ
δδδ
δδ
−−−
+≥−−
− ∑+
=
−+
1)ˆ)(1(
1)1(
''
'1 puupu F
m
F
mt
ttit
ttFm
for all t, or
pp mmt
ttit
tt ˆ)1()1('
'' δδδδ −≤− ∑
+
=
− for all t.
(2b) If i is a casual, },ˆmin{)1()1( 0'
'' ppp m
mt
ttit
tt δδδδ −≤− ∑+
=
− for all t.
Adding up these inequalities across all consumers, we have:
≤∆+− ∑+
=
− )()1( 0'
' πδδmt
tt
tt }],ˆmin{)1(ˆ[)1( 0ppqpqNm −+−δδ , or
∆+≥⋅−+⋅⋅−−
+ 001 }],ˆmin{)1(ˆ[1
)1( πδδδ ppqpqNm
m
.
From these results, we can derive the same conditions as in Proposition 1.
APPENDIX B: The General Model
We now offer a generalization of Proposition 1, which results in Proposition 2 below. We
maintain the assumption of a fixed, stable market environment, but now allow for a high degree
of heterogeneity among consumers and a positive marginal cost. It turns out that we only need to
characterize the following parameters for each individual consumer i:
=iδ i’s time discount factor,
=iv i’s valuation of S’s product less i’s net utility gain (before considering time discount)
from her most preferred outside option,
where no purchase (zero utility) is counted as an available outside option. For example, in our
standard model, for a fan, FFi uuv =−= 0 , since a fan’s most preferred outside option is no
purchase; for a casual, 00 )( ppuuv CCi =−−= . S is effectively a monopoly when consumer
valuations are defined according to iv ; moreover, S’s potential market is defined by the set of
consumers with 0≥iv . Denote as ),( δvG the cumulative distribution function of iv and iδ
among the consumers, so that, given a randomly chosen consumer i, =),( δvG the probability
that vvi ≤ and δδ ≤i . The total number of consumers is N.
Suppose that each product sold incurs a marginal cost c for S, and S’s optimal per period
profit maximizing price under FP is p̂ , so that its optimal per period profit under FP with
constant price is:
−=−= ∫∫
≥≥ pvp
pv
vdGcpNvdGcpN ),()(max),()ˆ(ˆ
0 δδπ .
Assume that c is not so high that S would prefer to exit the market if it operates under FP. Also,
as with other parameters, assume that c is constant across periods, so that p̂ must also be
constant across periods. Then, we have the following:
PROPOSITION 2. A PWYW strategy with threshold ≥π 0π and m punishment periods
admits consumer-side equilibria in which S always allows PWYW, iff:
∫∫≥
+
≥>
+ −−−≥−−−pv
mm
vp
mm vdGpvdGcvˆ
1
0ˆ
1 ),()]}1/()1([1{ˆ),(})]1/()1({[ δδδδδδδδ ,
and π satisfies:
00
1 ),(}}ˆ,min{)]1/()1({[ ππδδδδ ≥≥−⋅−−∫≥
+
v
mm vdGcpvN .
Further, for every such π , any member of this set is characterized by a payment scheme }{ ip
with which consumer i pays ip for S’s product in any period with PWYW, and which satisfies:
(i) }ˆ,min{)]1/()1([0 1 pvp imi
miii
+−−≤≤ δδδ for all i, and
(ii) 0ππ ≥=∑i
ip .
Proof of Proposition 2. The proof is very similar to that of Proposition 1 but requires more
general terminology. We shall only focus on PWYW equilibria with stationary payments, i.e.,
each consumer pays the same amount under PWYW across periods (although payments across
consumers can be different). Extension into non-stationary payments is very similar to previous
discussions in the context of the standard model too, and will not be repeated here, in the interest
of brevity.
First of all, as noted in the main text, consumer i would never buy from S, even under
PWYW (when she could pay nothing for S’s product), if 0<iv . This is because iv is the
difference between i’s valuation of S’s product ( i.e. the maximum net utility gain she can earn
from S’s product) and the net utility gain she can earn from purchasing her most preferred
outside option. Thus, if 0<iv , she still prefers to purchase her most preferred outside option
even when she can obtain S’s product for free. Conversely, if 0≥iv , i buys from S under
PWYW. Thus, overall, under PWYW, a consumer buys from S iff 0≥iv .
Consider a candidate PWYW equilibrium with specified per period profit ∆+= 0ππ
( 0≥∆ ), which is also the threshold that S uses for its credible threat. Consider a possible
payment scheme for this equilibrium under which consumer i always pays ip for S’s product in
a period; only a consumer i with 0≥iv would ever join this scheme. The payment scheme can
be represented by the set }{ ip . Similar to the corresponding portion of the proof in Proposition
1, we must have (with marginal cost c):
(1) ∆+=−∑≥
0}0:{
)( πivi
i cp ,
where we explicitly state in the equation the domain of consumers over which the summation
applies.
Now, for consumer i, let her valuation of S’s product be 0≥iu .1 Then, once S switches to
FP with posted price p̂ , i buys S’s product according to whether ( pui ˆ− ) (the net utility gain
from purchasing S’s product) is not less than ( ii vu − ) (the net utility gain from purchasing the
outside option). Hence once S switches to FP, i’s per period net utility gain must be
},ˆmin{ ii vpu − . Thus the incentive compatibility condition for i to adhere to the payment
scheme }{ ip when punishment lasts m periods is:
1 Since i’s net utility gain from her most preferred outside option must be non-negative, so that we effectively only need to consider consumers with vi ≥ 0 we can assume that ui ≥ 0 for all i without loss of generality; consumers with negative iu would definitely not buy from S and need not be considered as part of the market.
(2) i
iimii
iiii
mi vpu
upuδ
δδδδ
−−−
+≥−⋅−− +
1}),ˆmin{)(1(
)(1
1 1
, or },ˆmin{1
)1(1 im
i
mii
i vpp +−−
≤δδδ
.
Combining (1) and (2), we have (noting that ∫≥
−=pv
vdGcpNˆ
0 ),()ˆ( δπ ):
∫∫≥
+
≥>
+ −−−≥−−−pv
mm
vp
mm vdGpvdGcvˆ
1
0ˆ
1 ),()]}1/()1([1{ˆ),(})]1/()1({[ δδδδδδδδ .
These make up the necessary condition pertaining to Proposition 2. The upper bound for per
period payment, i.e., (i) in Proposition 2, is simply the results in (2). Constraint (ii) in
Proposition 2 is simply the condition (1) stated earlier in this proof.
For the “if” part of Proposition 2, suppose the condition
∫∫≥
+
≥>
+ −−−≥−−−pv
mm
vp
mm vdGpvdGcvˆ
1
0ˆ
1 ),()]}1/()1([1{ˆ),(})]1/()1({[ δδδδδδδδ
is fulfilled. Then, it is possible to choose π such that
00
1 ),(}}ˆ,min{)]1/()1({[ ππδδδδ ≥≥−⋅−−∫≥
+
v
mm vdGcpvN ,
as ∫≥
−=pv
vdGcpNˆ
0 ),()ˆ( δπ . With such a π , consider the following payment scheme:
}ˆ,min{)]1/()1([ 1 pvp imi
miii
+−−= δδδε for every consumer i, where
∫≥
+ −⋅−−=
0
1 ),(}}ˆ,min{)]1/()1({[v
mm vdGcpvN δδδδπε .
We must have 01 >≥ ε given our choice of π . Therefore, if the first condition in
Proposition 2 is fulfilled, then there exists π satisfying:
00
1 ),(}}ˆ,min{)]1/()1({[ ππδδδδ ≥≥−⋅−−∫≥
+
v
mm vdGcpvN ,
and for every such π , a PWYW strategy with threshold π and m punishment periods admits a
set of consumer-side equilibria (with at least one member as in the one example we just
discussed) according to which S always allows PWYW and makes profit π every period. The
payment scheme associated with the discussed example, moreover, certainly satisfies the
constraints (i) and (ii) in Proposition 2.
It is straightforward to verify that Proposition 2, when applied to the model in the main text,
yields Proposition 1. The insights and discussion after Proposition 1 also apply here.
As an end note to this section, we examine the impact of a change in c on the existence of
PWYW equilibria with higher profit than under perpetual FP. Intuitively, one might expect that
a higher marginal cost creates more loss with each low-payment transaction under PWYW, thus
making PWYW equilibria less profitable. But the optimal perpetual FP profit changes at the
same time, rendering comparison between equilibrium PWYW and perpetual FP profits less
straightforward. In more detail, first define the function:
∫∫≥
+
≥>
+ −−−−−−−≡pv
mm
vp
mm vdGpvdGcvcKˆ
1
0ˆ
1 ),()]}1/()1([1{ˆ),(})]1/()1({[)( δδδδδδδδ .
By Proposition 2, PWYW equilibria exist with higher profit than under perpetual FP iff
0)( ≥cK . Note that p̂ is non-decreasing in c (Tirole 1988, p. 66-67), so that we can divide our
discussion into two cases: 0/ˆ =∂∂ cp and 0/ˆ >∂∂ cp .
For the case 0/ˆ =∂∂ cp , it is obvious that 0/ <∂∂ cK , and so increasing marginal cost could
lead to PWYW equilibria with higher profit than under perpetual FP becoming non-existent.
For the case 0/ˆ >∂∂ cp , it is more convenient to calculate the derivative of K w.r.t. p̂ rather
than c, from which we obtain:
),ˆ(ˆ)ˆ(ˆ),()]}1/()1([1{),(ˆ
)ˆ()ˆ(ˆˆ ˆ
1
0ˆ
pppgpvdGvdGpcpcgpp
pK
pv
mm
vp
δδδδδδδ −+−−−−∂∂
−−=∂∂
∫∫≥
+
≥>
where ∫<≤
=10
),ˆ()ˆ(δ
δpdGpg and ∫<≤
+−−=10
1 ),ˆ()]1/()1([)ˆ(δ
δδδδδ pdGp mm .
Thus:
∫∫≥
+
≥>
−−−−∂∂
−−=∂∂
pv
mm
vp
vdGvdGpcpgcp
pK
ˆ
1
0ˆ
),()]}1/()1([1{),(ˆ
)ˆ()ˆ(ˆ
δδδδδ .
Hence an increase in c, which leads to an increase in p̂ , could lead to an increase in K if: (1)
)ˆ( pg is large, i.e., many more consumers are priced out with the increase in p̂ ; (2) pc ˆ/ ∂∂ is
small, i.e. cp ∂∂ /ˆ is large, or p̂ increases sharply with c ; (3)
∫≥
+−−−pv
mm vdGˆ
1 ),()]}1/()1([1{ δδδδ is small, i.e., demand is not high at price p̂ , and those who
buy from S at p̂ have high time discount factors. Then, counter-intuitively, an increase in
marginal cost could bring about the existence of PWYW equilibria with higher profit than under
perpetual FP.
APPENDIX C: PWYW Strategy with Probabilistic Punishment
We first prove a set of equilibrium results analogous to Proposition 1. As before, we shall
only focus on pure-strategy, stationary equilibria. Because of the increased complexity of the
model, break them down into one proposition and two corollaries:
PROPOSITION 3. Consider a PWYW strategy with probabilistic punishment with upper
threshold π , lower threshold π , and m punishment periods. If this strategy admits a consumer-
side equilibrium with which total payment is positive under PWYW, then S makes profit π every
period and continuously allows PWYW with certainty. Moreover, any such equilibrium is
characterized by a payment scheme }{ ip with which consumer i pays ip for S’s product in any
period with PWYW, and which satisfies:
(i) imm
i vp )]1/()1([0 1+−−≤≤ δδδ if imm v)]1/()1([ 1+−−≤− δδδππ ,
(ii) )()]1(/)1[(0 ππδδδ −−−−≤≤ mii vp if i
mm v)]1/()1([ 1+−−≥− δδδππ , and
(iii) π=∑i
ip ,
where pvi ˆ= when i is a fan and 0pvi = when i is a casual.
Proof of Proposition 3. Consider a (pure-strategy, stationary) equilibrium payment scheme }{ ip .
By definition, every ip must be a best response to all other consumers’ payments, given the
firm’s punishment strategy. Let ∑≠
− =ij
ii pλ . Denote as )( tΛρ the probability that S continues
PWYW in period t+1 given that it allows PWYW in period t and the sum of payments turns out
to be tΛ in that period. As described in the main text, 0)( =Λ tρ when π<Λ t , 1)( =Λ tρ
when π≥Λ t , and )/()()( πππρ −−Λ=Λ tt when ππ <Λ≤ t . Denote as )'( ipU the total
time-discounted utility of i if i pays 'ip whenever PWYW is allowed, given other consumers’
strategies. Suppose i is a fan. Then:
+
−−−
⋅+−+++−= −− )'()1(
)1)(ˆ()]'(1[)'()'()'()'( im
mF
iiiiiiFi pUpuppUppupU δδ
δλρλρδ ,
and so )'()1()1(
1)]'(1)[ˆ)(1(
')'( 1
iimm
iiFm
iF
i p
ppupu
pU−
+
−
+−−−−
+−−−+−
=λρδδδ
δλρδδ
, which can be rewritten as:
.)'()1()1(
'ˆ1
ˆ)'()1()1(
)]'()1()1)[(ˆ()]1()1()[ˆ()1)('(1
1
)'()1()1()]'(1)[ˆ)(1()1)('(
11)'(
1
1
11
1
iimm
iF
iimm
iimm
Fmm
FiF
iimm
iiFm
iFi
ppppu
pppupupu
pppupu
pU
−+
−+
−++
−+
−
+−−−−
+−−
=
+−−−+−−−−+−−−−+−−
−=
+−−−+−−−+−−
−=
λρδδδδ
λρδδδλρδδδδδδδ
δ
λρδδδλρδδδ
δ
When i is a casual, it can be similarly shown that:
)'()1()1('
1)'( 1
00
iimmiC
i ppppu
pU−
+ +−−−−
+−−
=λρδδδδ
.
Therefore, if we define iv such that pvi ˆ= when i is a fan and 0pvi = when i is a casual,
maximization of )'( ipU is equivalent to maximizing:
)'()1()1('
)'( 1ii
mmii
i ppv
pV−
+ +−−−−
=λρδδδ
.
If πλ ≥−i , 1)'( =+ −iip λρ for any non-negative 'ip , and i’s best response is to pay zero.
Therefore, the best response is positive only if πλ <−i and we shall focus on this case in the rest
of this proof. We need first to consider a number of different regimes of 'ip , find out the
optimal strategy in each regime, and then find out which of those is the overall best response. If
0≥− −iλπ , then there exists a regime iip −−≤≤ λπ'0 over which 0)'( =+ −iip λρ , and i’s
optimal strategy over that regime is zero payment. Next, over the regime 'ii p≤− −λπ ,
1)'( =+ −iip λρ and the optimal strategy over that regime is to pay i−− λπ . Lastly, over the
regime iii p −− −<<− λπλπ '},0max{ , )/()'()'( πππλλρ −−+=+ −− iiii pp , and so:
.)]}/()')(1([)1){((
))(1())(1(
)]}/()')(1([)1{()]/()1()'[()]/()')(1([)1(
')'(
21
1
21
1
πππλδδδπππλδδππδ
πππλδδδππδδπππλδδδ
−−+−−−−−+−+−−−
=
−−+−−−−−−+−−+−+−−
=∂
∂
−+
−+
−+
−+
iimm
iimm
iimm
miiii
mm
i
i
pv
ppvp
ppV
The sign of the above does not depend on 'ip , and so )'( ipV is monotonic over
iii p −− −<<− λπλπ '},0max{ . Thus the optimal strategy over this regime must be a limiting
corner solution, i.e. either },0max{ i−− λπ or i−− λπ . Combining this with previous
observations, the overall optimal strategy is either to pay i−− λπ or zero. This implies that, if an
equilibrium with total positive payment under PWYW exists, i.e. if some of }{ ip is positive,
then the total payment must be π . This confirms the condition (iii) π=∑i
ip in Proposition 3.
To find out the upper bounds in (i) and (ii) of Proposition 3, observe that the best response is
positive only if )0()( VV i ≥− −λπ , i.e.
)()1()1()1()1()(
11i
mmi
mmii vv
−++
−
−−−≥
−−−−−
λρδδδδδδλπ
, or
)]()1()1)[(()](1)[1( 1i
mmiii
m v −+
−− −−−−≥−− λρδδδλπλρδδ .
Recalling that the best response is positive only if πλ <−i , we only need to tackle two cases:
(1) πλ ≤−i so that 0)( =−iλρ . Then the above inequality becomes:
im
m
ii vp)1()1(
1+− −−
≤−=δδδλπ ;
(2) πλπ <≤ −i so that )/()()( πππλλρ −−= −− ii . Then the above inequality becomes:
)1())(1(
miii vpδδ
ππδλπ−
−−−≤−= − .
Observe that:
πδδδπ ≤
−−
− + im
m
v)1()1(
1 when im
m
v)1()1(
1+−−
≤−δδδππ , while
πδδ
ππδπ ≥
−
−−−−
)1())(1(
miv when im
m
v)1()1(
1+−−
≥−δδδππ .
This means that only one of either (1) or (2) applies to any consumer i. From these results, we
have the upper bounds (i) and (ii) in Proposition 3.
For the following two corollaries, recall that },max{ 00 pquN F=π is the optimal per period
profit under perpetual FP with constant price.
COROLLARY 1. Given a probabilistic punishment strategy with which 0ππ ≥ , PWYW-
sustaining equilibria exist iff 0pquF ≥ and one of the following is satisfied:
(1) 011)1( pm
m
+−−
≤−δδδππ and ])1([
1)1(
01 pqquN Fm
m
−+−−
≤ +δδδπ ;
(2) Fm
m
m
m
up)1()1(
)1()1(
101 ++ −−
≤−≤−−
δδδππ
δδδ and
01 )1()1()1()(
)1()1)(1( pqquq
N Fm
m
m −+−−
≤−−
−−+ +δ
δδππδδ
δπ ; or
(3) ππδδδ
−≤−−
+ Fm
m
u11)1( and 0)1()(
)1()1( pqqu
N Fm −+≤−−−
+ ππδδδπ .
Proof of Corollary 1. First consider 0pquF ≥ . Then Fi uv = for fans. The “only if” direction
for (1) to (3) follow immediately from summing up the payment upper bounds (i) and (ii) in
Proposition 3 across consumers under different conditions, and noting that the total payment
must be π by (iii). The “if” direction can be proved by construction in a similar way as with
Proposition 1, and is omitted here. Next, consider 0pquF < . Then 0pvi = for all consumers,
and summing up the upper bounds give the necessary conditions:
(1a) 011)1( pm
m
+−−
≤−δδδππ and 011
)1( pN m
m
+−−
≤δδδπ ;
(2a) ππδδδ
−≤−−
+ 011)1( pm
m
and 0)()1(
)1( pN m ≤−
−−
+ ππδδδπ .
Both imply that 00 ππ =< Np , leading to a contradiction.
COROLLARY 2. There exists a probabilistic punishment strategy with m punishment
periods with which (a) Fmm u)]1/()1([ 1+−−≤− δδδππ , (b) 0ππ ≥ , and (c) there exist PWYW-
sustaining equilibria, iff:
(1) 0pquF ≥ and
(2a) 1)1(
11
)1( 01 ≥
−+
−−
+F
m
m
qupq
δδδ .
There exists a probabilistic punishment strategy with m punishment periods with which (a)
Fmm u)]1/()1([ 1+−−≥− δδδππ ,(b) 0ππ ≥ , and (c) there exist PWYW-sustaining equilibria, iff:
(1) 0pquF ≥ and
(2b) 1)1(
11 0
1
≥−
⋅−
− +
F
m
upq
δδ .
Proof of Corollary 2. The condition (1) 0pquF ≥ is reproduced from Corollary 1. Conditions
(2a) and (2b) are proved with similar approaches and we shall only present that for (2b). Since:
Fmm u)]1/()1([ 1+−−≥− δδδππ ,
we know that (3) of Corollary 1 applies. Note that Fmm
F uu )]1/()1([ 1+−−>≥ δδδπ if
FNqu=≥ 0ππ . Therefore, given an upper threshold π , a choice of lower threshold exists that
satisfies the above constraint as well as the second inequality of (3), iff:
011 )1(1
1)1()1(
)1()1( pqquu
Nu
N FFmFm
m
m −+≤−−
+=−−
⋅−−
+ ++ δδπ
δδδ
δδδπ .
Incorporating the additional requirement that FNqu=≥ 0ππ produces (2b).
It is generally achievable that, in a PWYW equilibrium, every consumer finds paying
according to equilibrium to be strictly preferable to paying zero, i.e. for all i:
)()1()1()1()1()(
11i
mmi
mmii vv
−++
−
−−−>
−−−−−
λρδδδδδδλπ
.
This is true whenever the equilibrium payment is strictly lower than the applicable upper bound
for each consumer i as laid out in Proposition 3. Given this and the fact that )(⋅ρ is a continuous
function when ππ > , even if the sum of all other consumers’ payments consistently fall below
i−λ , as long as the difference is always sufficiently small, paying zero (i.e. following the baseline
equilibrium) is strictly dominated by maintaining equilibrium payment for i. Formally, define
'i−λ as the smallest possible sum of payments of all consumers except i in any period. Then the
above intuition can be captured by the following: there exists 0>ε such that, for all i,
⇒>−> −− 0'ii λλε)'()1()1()'()1()1(
)(11
imm
i
iimm
ii vv
−+
−−+
−
−−−>
+−−−−−−
λρδδδλλπρδδδλπ
.
The same cannot be said with the “step function” punishment, which is effectively the limit
ππ → , and with which )'( ii −− +− λλπρ drops from one to zero whenever 'ii −− − λλ increases
from zero to an arbitrarily small positive amount. Thus probabilistic punishment provides
additional stability to PWYW equilibria over the “step function” punishment.
APPENDIX D: Experimental Instructions for Strategy 1, Chat/No Suggestion Condition
Neutral Context Rich Context
INSTRUCTIONS THE ONLINE MUSIC GAME
INSTRUCTIONS
In this study you will make many decisions. Your payment at the end of the study will depend on your own decisions as well as the decisions of others you play with. During this study you will play the same game 20 times, and each game will consist of a number of rounds.
Description of the Game The game is played by 8 players. In each round, a player can choose one of three options: option S or option R or option N.
The game is played by 8 players. Two bands, the Playa band and the Quello band, upload a new song to their site every round of the game and allow people to listen to it online. In each round, a player can either listen to a Playa song or to a Quello song or to none.
Types of Players There are two types of players, who differ in how much they value each option. 1. Type Y players – To these players, option
S is equivalent to gaining 201 tokens (the experimental currency used in this study that will be later converted to real money); but options R and N are equivalent to gaining nothing i.e. 0 token.
2. Type Z players – To these players,
options S and R are equivalent to gaining 49 tokens, but option N is equivalent to gaining nothing i.e. 0 token.
There are two types of players, who differ in how much they like each band. 1. Fans of Playa – these people love Playa’s
music but not Quello’s. To them, listening to a Playa song is equivalent to gaining 201 tokens (the experimental currency used in this study that will be later converted to real money); but listening to a Quello song is worth nothing to them, and is in fact equivalent to gaining 0 token.
2. Casual Listeners – these people like
listening to music in general but do not prefer one band over the other. Listening to any song – be it Playa or Quello –
To sum up:
Type of player
Gain from selecting one option
S R N
Y 201 tokens 0 token 0
token
Z 49 tokens
49 tokens
0 token
Of the 8 players in the game, two are of Type Y and six are of Type Z
means the same to them, and is always equivalent to gaining 49 tokens.
To sum up:
Type of player
Gain from listening to a song of
Playa Quello Fan of Playa
201 tokens 0 token
Casual Listener 49 tokens 49 tokens
Of the 8 players in the game, two are Fans of Playa and six are Casual Listeners.
We will assign these roles to you and the other players before each game begins. Every player’s role will be fixed in all the rounds of the same game, but will be re-assigned from game to game. Each player will be of Type Y in 5 games and Type Z in 15 games.
Each player will be a Fan of Playa in 5 games and a Casual Listener in 15 games.
Payment To choose an option, you may need to pay. Different options have different payment schemes: • The payment scheme for option S. In the
first round of every game, anyone who chooses S can pay as he/she wishes. In other words, if you choose S you may pay nothing (0 token), or you may pay any number of tokens you wish; it is entirely up to you how much you pay for S. However, this scheme of “pay as you wish” for S will continue to the next round only if the total payment for S from all those who have chosen S in this round is at least 400 tokens; otherwise, S will cost 200 tokens in all future rounds of the game. In general, as long as the total payment
To listen to a song, you may need to pay. The two bands have different payment schemes: • Playa payment scheme. In the first round
of every game, Playa allows every listener to pay as he/she wishes. In other words, if you listen to a Playa song you may pay nothing (0 token), or you may pay any number of tokens you wish; it is entirely up to you how much you pay Playa for its song. However, Playa will continue to carry out this “pay as you wish” scheme only if it receives at least 400 tokens from all those who have listened to the Playa song in this round; otherwise, it will charge a fixed fee of 200 tokens per song in all future rounds of the game. In general, Playa’s management decides
for S is at least 400 tokens in a round, the payment for selecting S will continue to be “pay as you wish”; but once the total payment for S in a round is less than 400 tokens, the payment scheme for S will change and S will cost 200 tokens per round in all future rounds of the game.
• The payment scheme for option R. You
pay 48 tokens to choose R in a round, and this is the same in all rounds.
• The payment scheme for option N. You
pay nothing i.e. 0 token to choose N in a round, and this is the same in all rounds.
that, as long as the total payment that Playa receives from its listener(s) is at least 400 tokens in a round, Playa will continue to allow every listener to pay as he/she wishes; but once Playa receives less than 400 tokens in a round, it will change its payment scheme and will charge every listener a fixed fee of 200 tokens per round in all future rounds of the game.
• Quello payment scheme. Quello always
charges a fixed fee of 48 tokens per song per round.
How much does a player earn from choosing an option in a round?
How much does a player earn from listening to a song in a round?
Your (and every other player’s) earnings from choosing an option in a round are calculated as follows: Earnings from choosing an option = Gain from choosing the option – payment Choosing R If a Type Z player chooses R in a round, he/she gains 49 tokens, but has to pay 48 tokens. Thus his/her earnings in that round are equal to: Earnings = 49 tokens – 48 tokens = 1 token. So a Type Z player earns 1 token for choosing R in a round. If a Type Y player chooses R in a round, he/she gains 0 token, but has to pay 48 tokens. Thus his/her earnings in that round are equal to:
Your (and every other player’s) earnings from listening to a song in a round are calculated as follows: Earnings from listening to a song = Gain from listening to the song – payment Listening to a Quello song If a Casual Listener listens to a Quello song in a round, he/she gains 49 tokens from listening to the song, but has to pay 48 tokens for the song. Thus his/her earnings in that round are equal to: Earnings = 49 tokens – 48 tokens = 1 token. So a Casual Listener earns 1 token for listening to a Quello song in a round. If a Fan of Playa listens to a Quello song in a round, he/she gains 0 token from listening to the song, but has to pay 48 tokens for the song. Thus his/her earnings in that round are
Earnings = 0 token – 48 tokens = -48 tokens. So a Type Y player loses 48 tokens for choosing R in a round. To sum up:
Type of player Earnings from choosing R
Y -48 tokens
Z 1 token
Choosing S If a player chooses S in a round, and if S has a fixed cost of 200 tokens in that round, the calculation of earnings is similar to that for choosing R, but with a cost of 200 tokens instead of 48 tokens. But if the payment scheme for S is “pay as you wish” in that round, the payment and thus earnings of a player may vary from player to player depending on how much (if at all) each player pays. To sum up:
Type of player
Earnings from
choosing S in a round
under “pay as you wish” (supposing
player pays m tokens)
Earnings from
choosing S in a round under fixed
cost (of 200
tokens per round)
Y (201 – m) tokens
(201-200) = 1 token
Z (49 – m) tokens
(49-200) = -151 tokens
If a player chooses option N in a round,
equal to: Earnings = 0 token – 48 tokens = -48 tokens. So a Fan of Playa loses 48 tokens for listening to a Quello song in a round. To sum up:
Earnings from listening to a Quello song
Fan of Playa -48 tokens Casual
Listener 1 token
Listening to a Playa song If a player listens to a Playa song in a round, and if Playa charges a fixed fee of 200 tokens in that round, the calculation of earnings is similar to that for listening to a Quello song, but with a payment of 200 tokens instead of 48 tokens. But if Playa allows every listener to pay as he/she wishes in that round, the payment and thus earnings of a player may vary from player to player depending on how much (if at all) each player pays. To sum up:
Earnings from
listening to a Playa song in a round under “pay
as you wish” (supposing player pays m tokens)
Earnings from
listening to a Playa
song in a round under
fixed fee (of 200
tokens per round)
Fan of Playa
(201 – m) tokens
(201-200) = 1 token
Casual Listener
(49 – m) tokens
(49-200) = -151
tokens If a player does not listen to any song in a
his/her earnings in that round is 0 token.
round, his/her earnings in that round is 0 token.
How does the game continue after a round is finished? A round is finished when all players have made their decisions for that round. After that, the computer will randomly select whether the game will end or whether it will proceed to the next round. After each round, there is a 90% chance that the game will continue to the next round and a 10% chance that the game will end immediately.
The chat forum Before each game, there is a chat forum with limited duration through which players in the same group can send messages to each other. You are allowed to: 1. chat 3 minutes before each game in Games 1 to 6, 2. chat 2 minutes before each game in Games 7 to 13, and 3. chat 1 minute before each game in Games 14 to 20. 1.
Procedures You will enter all your decisions via the computer terminal in front of you. As each game begins, you will see on the computer screen whether you have been randomly assigned to be a Type Y or a Type Z player. Remember that: (1) there are two Type Y players and six Type Z players in every game; (2) every player’s role (Type Y/Type Z) will be fixed in all the rounds of the same game, but will be re-assigned from game to game; and (3) each player will be of Type Y in 5 games and Type Z in 15 games. At the beginning of each round, you will see a Decision Screen such as Decision Screens (1) to (4) (Please refer to the handout on your desk labeled “Decision Screens”.) A Decision Screen lists the following from top to bottom: • Whether you are a Type Y or a Type Z
player. • The number of the current game and
You will enter all your decisions via the computer terminal in front of you. As each game begins, you will see on the computer screen whether you have been randomly assigned to be a Fan of Playa or a Casual Listener. Remember that: (1) there are two Fans of Playa and six Casual Listeners in every game; (2) every player’s role (Fan of Playa/Casual Listener) will be fixed in all the rounds of the same game, but will be re-assigned from game to game; and (3) each player will be a Fan of Playa in 5 games and a Casual Listener in 15 games. At the beginning of each round, you will see a Decision Screen such as Decision Screens (1) to (4) (Please refer to the handout on your desk labeled “Decision Screens”.) A Decision Screen lists the following from top to bottom: • Whether you are a Fan of Playa or a
Casual Listener. • The number of the current game and
current round. • The payment scheme of S, R and N in the
current round. • Three buttons that correspond to the
decisions you can make in that round, i.e. (from left to right) (1) choose S; (2) choose N; (3) choose R.
• Under each button, you will see a column of numbers that include: (i) Your gain in tokens if you choose the
option labeled on the button; (ii) The cost you would have to pay for
your decision, except if your decision is to choose S under “pay as you wish” (see (iv));
(iii) Your earnings as a result of your decision, except if your decision is to choose S under “pay as you wish” (see (iv));
(iv) If the payment scheme for S is of “pay as you wish” in the current round, then, in the space under the “S” button where the cost would be stated for other decisions, there is a blank. You may enter in the blank any potential payment that you are considering. After that, you may click the “Calculate” button to calculate your earnings from choosing S with that payment. You may repeat this process for as many potential payments as you like.
To make a decision, click the button labeled with your choice – except that if the payment scheme for S is “pay as you wish” in the current round, and you decide to choose S, you will need to: (1) enter your payment in the blank space under the “S” button, (2) click the “Calculate” button to see what your earnings will be, and (3) click the “S” button. Players choose options simultaneously. After
current round. • The payment scheme of Playa and Quello
in the current round. • Three buttons that correspond to the
decisions you can make in that round, i.e. (from left to right) (1) listen to a Playa song; (2) listen to no song; (3) listen to a Quello song.
• Under each button, you will see a column of numbers that include: (v) Your gain in tokens if your decision
is as labeled on the button; (vi) The fee that you would have to pay
for your decision, except if your decision is to listen to a Playa song under “pay as you wish” (see (iv));
(vii) Your earnings as a result of your decision, except if your decision is to listen to a Playa song under “pay as you wish” (see (iv));
(viii) If Playa allows every listener to pay as he/she wishes in the current round, then, in the space under the “Playa” button where the fee would be stated for other decisions, there is a blank. You may enter in the blank any potential payment that you are considering. After that, you may click the “Calculate” button to calculate your earnings from listening to a Playa song with that payment. You may repeat this process for as many potential payments as you like.
To make a decision, click the button labeled with your choice – except that, if Playa allows every listener to pay as he/she wishes in the current round, and you decide to listen to a Playa song, you will need to: (1) enter your payment in the blank space under the “Playa” button, (2) click the “Calculate” button to see what your earnings will be, and (3) click the “Playa” button. Players make listening decisions
all players have made decisions in a round, a feedback screen will appear that shows: (1) every player’s decision (S/R/N), payment, and earnings in that round; (2) the total payment for each option in that round. If the payment scheme for S is “pay as you wish” in the current round, the feedback screen will also show: (a) whether the total payment for S reaches 400 tokens; and (b) whether the “pay as you wish” payment scheme for S will continue in the next round of the game, or will change to a fixed cost in all future rounds of the game.
simultaneously and without communication with each other. After all players have made decisions in a round, a feedback screen will appear that shows: (1) every player’s decision (Playa/Quello/Neither), payment, and earnings in that round; (2) the total payment received by each band in that round. If Playa allows every listener to pay as he/she wishes in the current round, the feedback screen will also show: (a) whether the total payment received by Playa reaches 400 tokens; and (b) whether Playa will continue to allow every listener to pay as he/she wishes in the next round of the game, or will charge a fixed fee in all future rounds of the game.
Afterwards, the computer will select whether the game will proceed to the next round or will end. Remember that there is a 90% chance that the game will proceed to the next round, and a 10% chance that it will end. Once a game is ended, the next game will begin – unless you have already come to the last (20th) game, after which the study will be finished. You will play 20 games with the same group of 8 players (including yourself).
Payment
After all 20 games are finished, we will choose 5 games at random and pay you your total earnings from all the rounds in those games at a rate of HK$1 = 10 tokens. If you have any questions, please raise your hand and the study coordinator will come to speak to you. If you wish to participate in this study, please sign the consent form. Afterwards, when you are ready to start, please click the “START” button on the screen. You will then begin to play 1 practice game to familiarize yourself with the study; the practice game will not be chosen for calculation of your final payment. After the practice game is finished, you will play the 20 games of the study. Please wait patiently until all other players are ready to start.
APPENDIX E: Decision Screens (Rich Context) (1) A Fan of Playa’s Decision Screen when Playa carries out “pay as you wish”
(2) A Fan of Playa’s Decision Screen when Playa charges a fixed fee
(3) A Casual Listener’s Decision Screen when Playa carries out “pay as you wish”
(4) A Casual Listener’s Decision Screen when Playa charges a fixed fee
