Efficiency and the Redistribution of

WelfareMilan Vojnovic

Microsoft ResearchCambridge, UK

Joint work with Vasilis Syrgkanis and Yoram Bachrach

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Contribution Incentives• Rewards for contributions• Credits• Social gratitude• Monetary incentives

• Online services• Ex. Quora, Stackoverflow, Yahoo! Answers

• Other• Scientific authorship• Projects in firms

3

Que

stion

Topi

c

Site

4

Another Example: Scientific Co-Authorship

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Some Observations• User contributions create value• Ex. quality of the content, popularity of the generated content

• Value is redistributed across users• Ex. Credits, attention, monetary payments

• Implicit and explicit signalling of individual contributions• Ex. User profile page, rating scores, etc• Ex. Wikipedia – not in an article, but by side means [Forte and Bruckman]• Ex. Author order on a scientific publication

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How efficient are simple local value sharing schemes with respect to social welfare of the society as a whole?

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Outline• Game Theoretic Framework

• Efficiency of Monotone Games under a Vickery Condition

• Efficiency of Equal and Proportional Sharing

• Production Costs

• Conclusion

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Utility Sharing GameUSG():• : set of players

• : strategy space,

• : utility of a player,

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Project Contribution Games

Special: total value functions

1

2

i

n

1

2

j

m

Share of value

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Monotone Games• A game is said to be monotone if for every player and every

• It is strongly monotone, if for every player and every :

, for every

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Importance of Monotonicity• ,

• Nash equilibrium condition

• Efficiency =

0 1

1

𝑥

𝑣 (𝑥)concave,

𝑏∗

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Vickery Condition• A game satisfies Vickery condition if for every player and :

• It satisfies k-approximate Vickery condition if for every player and :

]

Rewarded at least one’s marginal contribution

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Local Value Sharing• A project value sharing is said to be local if the value of the project is

shared according to a function of the investments to this project:

, for every and

• Equal value sharing:

• Proportional value sharing:

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DBLP database• 2,132,763 papers• 1,231,667 distinct authors • 7,147,474 authors

Scientific Co-Authorships

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Scientific Co-Authorship (cont’d)

o random

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Solution Concepts & Efficiency• Nash Equilibrium (NE)• Unilateral deviations

• Strong Nash Equilibrium (SNE)• All possible coalitional deviations

• Bayes Nash Equilibrium (BNE)• Incomplete information game

• Efficiency• Worst case ratio of social welfare in an equilibrium and optimal social welfare

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Outline• Game Theoretic Framework

• Efficiency of Monotone Games under a Vickery Condition

• Efficiency of Equal and Proportional Sharing

• Production Costs

• Conclusion

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Efficiency in Strong Nash EquilibriumTheorem

• Any SNE of a monotone game that satisfies the Vickery condition achieves at least ½ of the optimal social welfare

• If the game satisfies the -approximate Vickery condition, then any SNE achieves at least of the optimal social welfare

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Efficiency in Nash EquilibriumTheorem

• Suppose that the following conditions hold:1) -approximate Vickery condition2) Strategy space of each player is a subset of some vector space3) Social welfare satisfies the diminishing marginal property

• Then, any NE achieves at least 1/() of the optimal social welfare

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Local Vickery Condition• A value sharing of a project is said to satisfy local k-approximate Vickery

condition if

• If value sharing of all projects is locally k-approximate Vickery, then the value sharing is k-approximate Vickery

• Local k-approximate Vickery condition

}

𝜕𝑖𝑣 𝑗 (𝒃 𝑗)

degree of substitutability

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Degree of Substitutability

• If value functions satisfy diminishing returns property, then

• If , then each player is quintessential to producing any value, i.e. , for every

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Degree of Substitutability (cont’d)• Efficiency =

• If , then any local value sharing cannot guarantee a social welfare that is 1/ of the optimum social welfare

1

2

𝑛

1

2

𝑛−1

⋮𝑣1 (𝒃)=𝑛∏

𝑖𝐼 (𝑏𝑖 ,1=1)

𝑣2 (𝒃 )=(1+𝜖) 𝐼 (𝑏𝑛 , 2=1)

Budget 1

}

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Outline• Game Theoretic Framework

• Efficiency of Monotone Games under a Vickery Condition

• Efficiency of Equal and Proportional Sharing

• Production Costs

• Conclusion

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Equal Sharing• Suppose that project value functions are monotone, then equal

sharing satisfies the -approximate Vickery condition

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Proportional Sharing

• Suppose that project value functions are functions of the total effort, increasing, concave, and

Then, proportional value sharing satisfies the Vickery condition

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Proof Sketch• concave and , for every

• Take and to obtain

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Local Smoothness• A utility maximization game is -smooth iff for every and :

• A utility maximization game is locally -smooth iff with respect with respect to at which are continuously differentiable:

where

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Efficiency of Smooth Games• If a utility sharing game is locally ()-smooth with respect to a strategy

profile then utility functions are continuously differentiable at every Nash equilibrium , then

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Sufficient Condition for Smoothness• are concave functions of total effort, , and are continuously

differentiable • Proportional sharing of value • For all strategy profiles and and ,

Then, the game is locally -smooth with respect to

if , else

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Efficiency by Smoothness:Fractional Exponent Functions

• Suppose that , and

• Then, proportional sharing achieves at least of the optimal social welfare

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Efficiency by Smoothness:Exponential Value Functions• Suppose , and

• Then, proportional value sharing achieves at least of the optimal social welfare

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Tight Example

• is a Nash equilibrium where each player focuses all effort his effort on project 1

1

2

𝑖

𝑛

1

2

𝑛

𝑛−1}𝑣1 (𝑥 )=1−𝑒−𝛼𝑥

𝑣2 (𝑥 )=𝑞 (1−𝑒−𝛽𝑥 )

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Tight Example (cont’d)• Nash equilibrium:

• Social optimum:

(players invest in distinct projects)

𝑢(𝒃)𝑢(𝒃∗)

→𝛼→∞,𝛽→ 0𝑛2

2𝑛2−2𝑛+1, large

1

2

𝑖

𝑛

1

2

𝑛

𝑣1

𝑣2}𝑛−1

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Efficiency and Incomplete Information• Proportional sharing with respect to the observed contribution

• Concave value functions of the total contribution

• Abilities are private information

Then the game is universally -smooth, hence, in a Bayes Nash equilibrium, the expected social welfare is at least ½ of the expected optimum social welfare

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Universal Smoothness• Game • Value function • Game is -smooth with respect to the function if

for all types and and every outcome that is feasible under

[Roughgarden 2012, Syrgkanis 2012]

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Efficiency under Universal Smoothness• Efficiency

• If a game is -smooth with respect to an optimal choice function then the expected social welfare is at least of the optimal social welfare

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Outline• Game Theoretic Framework

• Efficiency of Monotone Games under a Vickery Condition

• Efficiency of Equal and Proportional Sharing

• Production Costs

• Conclusion

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Production Costs• Payoff for a player: • Social welfare , total value

production cost

∞

00

𝑐 𝑖 (𝑥 )

𝑥 Budget

constraint(earlier slides)

00

𝑐 𝑖 (𝑥 )

𝑥 Constant

marginal cost

00

𝑐 𝑖 (𝑥 )

𝑥 A convex increasing function

Examples

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Elasticity• Def. the elasticity of a function at is defined by

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Efficiency• Suppose that production cost functions are of elasticity at least and

the value functions are of elasticity at most

• If is any pure Nash equilibrium and is socially optimal, then

Moreover

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Efficiency (cont’d)

• Constant marginal cost of production is a worst case• But this is a special case: for any production cost functions with a

strictly positive elasticity, the efficiency is a constant independent of the number of players • Budget constraints are a best case

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Conclusion• When the wealth is redistributed so that each contributor gets at least his

marginal contribution locally at each project, the efficiency is at least ½

• The degree of complementarity of player’s contributions plays a key role: the more complementary the worse

• Simple local value sharing• Equal sharing: the efficiency is at least 1/k, where k is the maximum number of

participants in a project• Proportional sharing: guarantees the efficiency of at least ½ for any concave project value

functions of the total contribution

• Production costs play a major function: the case of linear production costs is a special case for which the inefficiency can be arbitrarily small; at least a positive constant for any convex cost function of strictly positive elasticity