In 43 states in the USA, several cabinet posts, such as Attorney General, Secretary of State, Insurance Commissioner and Agriculture Commissioner, are elective offices. In 32 states, both the Secretary of State and the Treasurer are elective offices. Fifteen states fill four such posts through direct election and
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eighteen states have five or more such posts, l While the governor and lieutenant governor generally form a single ticket (as with electors for President and Vice-President), voters elect the other state cabinet members in individual elec- tions.
How will a political party choose which individuals to run for each of the different cabinet offices? With the exception of the Attorney General (who must be a member of the bar), cabinet officers frequently do not have considerable specific professional expertise prior to taking office. Therefore, a politician considering a run for a state cabinet office faces a choice of which position to seek. In many states, the cabinet as a whole has executive authority on certain issues. In such states, each political party wishes to fill as many cabinet seats as possible. How does a party want to form its slate to run against the rival party's slate? If voters' rankings of pairs of candidates is independent of the cabinet office for which they are running, we show that each party's equilibrium strategy is to assign candidates to cabinet offices randomly.
Similar decision problems arise for coaches of team sports, such as tennis, when the outcome depends on the results of multiple individual matches. Victory depends only on wins and losses in individual matches and how badly one side loses an individual match is irrelevant. Each team's coach has to choose which player (or players in doubles) to use against each of the opponent's players. Coaches submit a list of players for the matches. In principle, these lists are a ranking of the players in order of ability, No. 1, No. 2 and so on. In the match, each player plays the corresponding player on the opposing team's list. It will generally be difficult to enforce sincerely ranked assignments by coaches because players have streaks and player ability may not be transitive, among other reasons. What advice should a game theorist give a tennis coach on how to increase his chances of winning the team match? Should one try to match the best against the best or should one 'tank' a match (that is, match up one's worst player against the opponent's star player) to gain an advantage in other matches?
If opposing coaches maximize the probability of winning the team match, then they are playing a zero-sum game. No matter what the probabilities of any player defeating each opponent, an equilibrium strategy is to randomize completely the ordering of one's players, each player has an equal probability of playing in the No. 1 match, No. 2 match, etc. Given transitivity conditions which imply an unambiguous ranking of players, matching up the best against the best and so forth down the list is an optimal pure strategy for both teams only in the exceptional case when all matchings are equilibrium ones. Such cases occur only over trivial portions of parameter space and are, therefore, improbable. A sincere ranking of players is almost never in at least one coach's interest.
I These data are from 'The Book of the States 1994-1995', published by the Council of State Governments.
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Who benefits from the strategic 'insincere' ranking? A naive answer would be the weaker team when each player on the stronger team has a better than even chance of defeating the corresponding player (or every player) on the weaker team. We show, however, that the stronger team can gain in moving to the random assignment equilibrium.
All the latter points concerning team tennis matches have analogous interpreta- tions in the state cabinet officer problem. However, the political setting presents additional complications. The first is that parties may use primaries to select the members of the slate. But the assignment problem we describe below is still relevant. A party wants to select its most competitive set of candidates for its slate, i.e. it wants to avoid having its strongest candidates run in primaries against one another for the same spot on the party slate. If a party uses some mechanism, formal or informal, to resolve this internal coordination game in forming its ticket, it is effectively playing the assignment game against the rival party. A second complication is the presence of incumbents, who presumably do not try to run for different offices (again, unless the offices differ greatly in prestige). Then, the parties are playing the assignment game with respect to only the set of open seats, and our analysis applies.
Many other political contests conform to our problem. Many cities indepen- dently elect officers other than the mayor (comptroller and city council president in New York City, for example). In November 1992, two elections for open U.S. Senate seats in California took place. Some cities have several at-large council seats; if there are partisan contests for the individual seats, the two parties are playing our assignment game. In Section 5, we discuss how our model applies to Parliamentary elections in the absence of residency requirements.
We analyze the formal model in the context of our tennis example; interpreta- tions for the political analogue are obvious or discussed. Other examples of the problem appear at various points in the paper. Section 2 introduces the formal model. We solve for the equilibria and discuss their properties in Section 3. Section 4 considers some extensions to the model. Section 5 contains our conclusions.
2. The model
For expositional simplicity, we consider the tennis example with only singles matches. Two teams, A and B, each have n players, numbered 1 to n. If players can be unambiguously ranked according to ability (as defined below), let lower numbers correspond to better players. Our main results, Propositions 1 and 2, are independent of whether such a ranking exists, although our third proposition concerns cases with unambiguous player rankings. A match consists of each of the n players from team A playing an individual match against a different player from
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team B. The payoff function is discussed below, but note that it will depend only on the number of victories in the n individual matches. Let rrij ~ [0, 1 ] denote the probability that player i on team A defeats player j on team B, i, j = 1, 2 . . . . . n. (When it is clear by context, we dispense with presenting ranges of subscripts/su- perscripts.) Let 7r denote an n2-vector of values (7riy) which is an element of the
# \
parameter space H = ~ 7r ~ < 7rij < 1 ~.
A pure strategy is an assignment of a team's n players to the n slots (courts in tennis matches) where the individual matches will be played. A pure strategy is a permutation of the consecutively numbered players over the fixed n slots. Since each permutation is a feasible pure strategy, there are n! pure strategies.
A pair of pure strategies results in a 'match-up,' defined to be a unique pairing of the two team's n players. A particular match-up is independent of the slots in which the individual matches occur; given two strategies and the resulting match-up, one could permute the pairs o f players across slots without changing the match-up. Only the resulting match-up is relevant for the payoffs. For example, one can generate the complete set of match-ups by playing all of one team's pure strategies against the other team's sincere strategy when the latter exists. Henceforth, we refer to the pure strategy that assigns player i to slot i as the sincere assignment, although its 'sincere' connotation is meaningful only over the restricted region of H discussed below. Hence, there are also n! match-ups and at most n! distinct payoff pairs, although the normal-form matrix has (n[) 2 cells. More generally, the set of match-ups can be generated by arbitrarily fixing either team's pure strategy, and playing all the other team's pure strategies against it.
Table 1 shows the match-ups that pure strategy pairs yield in the case of two-player teams, where m 1 indicates team A's player i plays team B's player i and m 2 denotes the alternative match-up. A key property of this game is that, generally, each row and column of such a matrix is a permutation of the set of match-ups, (m l, m 2 . . . . . ran!).
Teams maximize their probabilities of winning the overall match, a function of the match-up (or match-ups when mixed strategies are played). This is, of course, a zero-sum game. Let pY(mi), i = 1, 2 . . . . . n!, denote team j ' s probability of winning given match-up m i, and v equal the total number of team A's victories.
Table 1 Match-ups for n = 2 Team A pure strategies Team B pure strategies
(1, 2) (2, 1)
(1, 2) m I m 2 ( 2 , 1) m 2 m I
J. Hamilton, R.E. Romano / European Journal of Political Economy 14 (1998) 101-114 105
Then:
p A ( m i ) =Pr[v > n/2[rni] + / 3 P r [ v = n / 2 [ m , ] and
p a ( m i ) -- 1 - - p A ( m i ) , (1)
)where/3 ~ [0, 1] equals the probability that team A wins after a tie in the initial set of matches (and 1 - / 3 is the probability that team B wins the tie breaker). The tie breaker might consist of a single match between one player from each team with an endogenous value for/3 that both teams anticipate. Alternatively, a /3 of 0.5 could reflect a coin toss or a one-half weight given to ties in league standings. Our results apply to other natural payoff functions as well, for example, the expected number of victories.
Let s j = (0i, 0~ . . . . . 0,J!) denote a mixed strategy of team j, j = A, B, where 0 / E [0, 1] equals the probability of playing pure strategy i (the numbering of team j ' s n! pure strategies is arbitrary) and ET: 10 /= 1. Let q J = [ 1 / n ! , 1 /n! , . . . . l / n ! ] denote team j ' s equal-probabilities strategy, having j use each assignment permutation with equal probability. Letting ri(s A, s B) denote the probability that match-up i occurs, PJ(s A, B _ n~ i A S ) = F_,i=lr (s , sB)p f fmi ) denotes team j ' s payoff. Finally, /" denotes the n! × n! matrix of team A's payoffs, found by replacing ra i in the matrix of match-ups (e.g. Table 1) with Pn(mi) .
Other sports and parlor-game examples include team chess, karate, and table tennis matches and pitching match-ups in baseball double headers z In some examples, the juggling of line-ups is feasible only for a subset of players. Suppose that the top m players on each tennis team must play on their respective courts 1 through m, but there is freedom in assigning the remaining n - m (weaker) players. This may arise if relatively strong players are publicly ranked and such rankings are used to limit strategic play. A pure strategy then consists of the required assignment of the top m players and an assignment of the remaining n - m players.
3. Properties of the equilibria
The game studied here is zero sum and finite. An equilibrium, possibly in mixed strategies, exists for any finite game. The distinctive feature of the present game is that each pure strategy confronts the opponent with exactly the same set of feasible payoffs. We refer to this property as pure strategy equivalence (PSE).
2 The presumption is that it is too cosily for either team to reassign its two available pitchers once the initial line-ups have been submitted. For example, the away team, which pitches second, cannot yank their pitcher in the fwst game after he has pitched to the first batter (a requirement), substitute the other pitcher, and use the first one in the second game.
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Each row and column of F contains the same elements. (PSE)
It is PSE that causes the zero-sum games studied here to have the properties shown below.
Our first proposition demonstrates that pure strategy equilibria exist only under very stringent conditions.
Proposition 1. A pure strategy equilibrium exists if and only if pa(m 1) = pA(m 2) = . . . =pA(mn!), in which case all strategy pairs, mixed and pure, are equilibrium strategy pairs.
Proof. Necessity can be shown by contradiction. Suppose (s A', s a') is an equilibrium pair of pure strategies, resulting in match-up m i. Assume pA(mj) pA(m i) for some mj. Then we obtain either the contradiction: (a) pA(mj) < pA(mi) and s a' is not a best reply to sX; or (b) pA(mj) > pA(m i) and s A' is not a best reply to s B'. The latter contradiction results since, by PSE, team A has a strategy that generates match up mj played against s w. Contradiction (a) arises since, by PSE, team B has a strategy that generates match up mj played against s x, with a higher payoff to team B because the game is zero sum. Hence, a pure strategy equilibrium does not exist unless: pA(m 1) = pA(m 2) = . . . = pA(m,!).
The condition pA(m 1) = pA(m 2) = . . . = pA(mn!) implies that team j is indif- ferent across strategy responses to any strategy choice of team i. Hence, all strategy pairs, pure and mixed, form an equilibrium. []
For any game with pure strategy equivalence, playing every pure strategy with equal probability is always an equilibrium.
Proposition 2. Equal probability mixed strategies played by both players, (s A, s a) = (qA, qB) always constitute an equilibrium. Generically, it is the unique equilibrium over the parameter space I-I.
Proof. Let s a = qB. Then, by PSE, A is indifferent among all its pure strategies and convex combinations of them, including qa. Similarly, if s A= qa, B is indifferent among all its pure strategies and convex combinations of them. Hence (qa, qB) is always an equilibrium strategy pair.
Genericity is shown by contradiction. Suppose B has distinct equilibrium strategies, s~ and s~. (An analogous argument holds for A.) Since the game is zero sum, equilibrium strategies are interchangeable (see, for example, Friedman, 1990, Theorem 3.2, p. 78). Then qA is a best response to s~ or s~. Since qA is completely mixed, playing any pure strategy against Sl B or s~ also yields A the same pay-off, i.e. Fs~ = Fs~. Distinctness of s A and s~ then implies F is singular. But nonsingularity of F is generic with respect to the space of payoffs,
J. Hamilton, R.E. Romano / European Journal of Political Economy 14 (1998) 101-114 107
so then the equilibrium is generically unique in payoffs 3. Since F is everywhere a locally unique mapping from H , generic uniqueness with respect to the parameter space follows. []
Suppose team B were to play a pure strategy. By PSE, team A could then obtain its most desired match-up, which is team B ' s least desired (since the game is zero sum). Similarly, should team B play a mixed strategy other than equal probabilities, team A could obtain its most desired match-up with probability greater than 1/n! . When a team mixes with equal probabilities, it provides its rival with minimal opportunity to obtain its most desired match-up. Since the game is zero sum, an equilibrium is also the minimax solution to the game. Equilibria of the game will have the same payoffs 4. Even in the rare cases of multiple equilibria, both teams fare no better in any other equilibrium than with completely random assignments s.
Generic uniqueness is easily illustrated for the case of two-player teams. Multiple equilibria requires p g ( m 1) = pA(m2). Using Eq. (1), p A ( m i) = Pr[v =
21rn i] + / 3 P r [ v -- l lmi] = 71"11'/1"22 -{- ~[orll(1 - - 71"22 ) "at- or22(1 - - o r l l ) ] • Likewise calculating p A ( m 2) and equating the payoffs yields the manifold of multiple equilibria:
( 1 - - 2 j ~ ) o r l l ' / r 2 2 + ]3('/ ' / '11 + 77"22 ] = (1 - 2/3)or,2or21 + fl(or,2 + or2,)-
(2)
If fl = 0.5 (e.g. a coin toss settles ties), then the manifold is a four-dimensional hyperplane in R 4 implying generic uniqueness. F o r / 3 4= 0.5, the first-order effect of a small deviation from or satisfying Eq. (2) is given by 6:
- - [ ( 1 -- 2/3)or2, + ~ ] A q r l 2 - [ ( 1 - - 2/3)or,2 + / 3 ] Aor2,.
For the solution to Eq. (2) to continue to hold for all small deviations in a neighborhood, the terms in brackets must vanish. However, for 0 < %~ < 1 and 13 ___ 1, these terms never vanish.
3 Bohnenblust et al. (1950, Theorem 3, p. 56) have shown generic uniqueness with respect to the pay-off space more generally for all discrete, two-person, zero-sum games.
4 Pay-off invariance for zero-sum games is formally proved in Friedman (1990), for example. Pure strategy equivalence can occur in games that are not zero sum, such as a special Battle of the
Sexes with equal payoffs to the two players in both the coordination and the coordination-failure outcomes. The proof that equal probability mixing is always an equilibrium under PSE does not require the game to be zero-sum. In the special Battle of the Sexes, equal probability mixing is an equilibrium, but other equilibria exist with higher payoffs. The payoffs from equal probability mixing are not a compelling prediction.
6 Genericity also holds for reasonable cases where fl is endogenous ( 13 =/3(~r)); for example a match between players selected from each team with equal probabilities breaks the tie.
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In tennis matches, the slots for players are ordered and there is often an expectation that a coach assigns the best player to the first slot, the second-best player to the second slot and so forth. It is not necessarily the case that a team can order its players independently of the opponent. However, when it is possible, we can compare the equilibrium outcome to the outcome where each team's best player faces the rival's best player, each team's second-best player faces the rival's second-best player and so forth.
There is an unambiguous ordering (UO) of player 's abilities if, for k = 2, 3, . . . , n:
1rij >_ Irkj for all i < k and for all j . (UO)
Under UO, player i is at least weakly stronger than player k and stronger if the inequality is strict for some j. We did not require UO above, although all our results continue to hold with it. Ignoring the possibility of equalities in parameters for simplicity, UO in the two-player case implies
7r12 > 1rll > 7r21 and 7r12 > Ir22 > 7r21 (3)
Many coaches share a belief that only weaker teams will manipulate assignments to gain an advantage. Perhaps surprisingly, either team may benefit from strategic assignments.
Proposition 3. An unambiguously stronger team can receive a higher or lower payoff in the Nash equilibrium than i f both teams make sincere assignments.
Proof. It is sufficient to consider the case where n = 2. Team A is unambigu- ously stronger given Eq. (3) and ~2t > 0.5 7. Let fl = 0.5 in the pay off function. Straightforward calculations imply the equilibrium outcome is preferred by team A over the pair of sincere assignments if 1.5(~12 + zr21 - 7rll - 7r22) + *rHTr22 - 7rl2zr2~ > 0. If 1rll = 7r22 = 0.9 and zr~2 = 0.96, then team A prefers the equilib- rium outcome to sincere assignments when zr2~ > 0.835. The opposite preference holds if ~'21 < 0.835. Hence, Eq. (3) and ~'21 > 0.5 are consistent with either pay off being higher. []
Proposition 2 implies one of the teams will (almos0 always want to deviate from sincere assignments. Proposition 3 implies that the team that gains from the move from sincere assignments to the random equilibrium assignments can be either the stronger or weaker one. We have found some empirical support for our prediction of strategic play. All the tennis coaches that we interviewed, college and high school, indicated that assignments are sometimes other than sincere. Tennis coaches refer to this as 'stacking.' However, we do not contend that the strategic assignments are as pervasive as our model predicts, as we discuss in Section 4.
7 A weaker definition of a stronger team would impose only wll > 0.5 and ~'22 > 0.5. The result still holds in this case.
J. Hamilton, R.E. Romano / European Journal of Political Economy 14 (1998) 101-114 109
4. Variations of the game
The evidence we found from sports coaches indicates that strategic play occurs frequently, but not to the point of complete randomization. A possible reason is that our simple model excludes important features of the true game. The same likely holds true for the political games we consider.
The payoff functions in Section 2 weighted all individual contests equally. For slates of state cabinet officers, it is plausible that parties value winning some offices more than others. The Attorney General's duties are more politically influential than those of the state Treasurer, so both parties probably care more about winning the contest for Attorney General. Then the appropriate payoff function would be:
y . Pr[win Attorney General] + (1 - y ) . Pr[win Treasurer], (4)
where 3' ~ (½, 1). The assignment game is still constant sum, but for 3' > ½, it no longer satisfies PSE. A pure strategy equilibrium with insincere assignments by one side may well arise for large differences in the importance of the different cabinet posts.
Assume Eq. (3) holds, so that we can speak of having better candidates competing for the more valued position (the Attorney General). Without signifi- cant loss of generality, assume that sincere assignments are not an equilibrium
1 with 7 = ~ and that party A has the incentive to deviate from sincerity:
~rl2 + ~r21 - ~rl~ - ~'22 > 0 (5)
Given Eqs. (3) and (5), sincere assignments by both teams is an equilibrium if and only if
71"12 - - ' / ' g22 3'> (6)
- 22) + -
Note that Eqs. (3) and (5) imply that the RHS of F_x 1. (6) exceeds ½. It is straightforward to confirm that neither party would deviate given Eq. (6), but if it fails to hold, party A would deviate from sincere assignments. Thus, Eq. (6) is necessary and sufficient for sincere assignments to be an equilibrium. Intuitively, with sufficient weight placed on the more important contest, each party will assign their better candidate there.
If Eq. (6) fails, then there may exist a pure strategy equilibrium with party B sincere and party A insincere, or a mixed strategy equilibrium. The following example illustrates all these possibilities. Suppose 1rll = 0.5; 7r12 = 0.75; qr21 = 0.45 and let ~r22 and 3' be parameters. The restriction (Eq. (3)) implies that ~r22 ~ (0.45, 0.7). Fig. 1 depicts the various equilibria that arise s. Note that we obtain a pure strategy equilibrium with only party A being insincere. It is never an
.833
.6
.5
ill
Mixed St ra tegy ~ / / / / / / S / / A ~ Equ i l ib r ium
J. Hamilton, R.E. Romano / European Journal of Political Economy 14 (1998) 101-114
Pure S t ra tegy Equi l ib r ium: Bo th Part ies S incere
11o
Y
1
x4~ d ', I
.45 .5 .7
Fig. 1. ( 0 / = Pr{Party j sincere}, j = A, B).
x 22
equi l ib r ium for B to use an ins incere pure strategy. The equ i l ib r ium is un ique except a long the boundar ies o f the lens (see s).
In this example , the re la t ive ly s t ronger par ty A some t imes uses ins incere ass ignments in an a t tempt to win both seats, whi le the w e a k e r par ty runs its bes t candida te for the more va lued seat. F o r o ther pa ramete r values , a w e a k e r par ty uses ins incere ass ignments , running its mos t appea l ing candida te for a less
s The equation of the lower boundary of the lens in Fig. 1 is "~(¢r~) = (¢r12 - "n'll)/(~'12 -- wll + Ir22 --~r2t)= 0.25/(w22--0.2). The equation of the upper boundary of the lens is $(¢r:2)= (¢rt2- ¢r22)/(cr12 - ¢r22 + Iril - ~'21) = (0.75 - ¢r22)/(0.8 - ¢r22). Along ~(w22) there are multiple equilib- ria where party A is insincere and party B is sincere with probability 0~ ~[0B(7, w22), 1] where _~ =(0.87-7wzz-0.05) / (0 .7-¢r22) . Along ~/(w22) but below its intersection with ~,, there are multiple equilibria with party B sincere and party A mixing with 0 h ~ [0, 1]. Along ~ but above the intersection with ~ there are multiple equilibria with party B sincere and party A sincere with probability 0 h ~ [~(7, ¢r22), 1] where _~ = (0.27,/- 7¢r22 -0.25)/(0.7 - ~'22).
J. Hamilton, R.E. Romano / European Journal of Political Economy 14 (1998) 101-114 111
important seat, in an attempt to guarantee itself at least one seat. The stronger party can continue to find it desirable to adopt sincere assignments.
A strong assumption of the basic model is that the probability that an individual defeats a particular individual on the other side is independent of the 'location' of the contest. In tennis matches, it is reasonable to presume that the outcome does not depend on which court a match is played on. In the state cabinet officer problem, the electorate is the same for all races, and it seems plausible that this holds true in this case as well. In multiple Parliamentary elections by districts, the outcome of a race say between a Conservative and a Labor candidate would be influenced by whether they compete in a district that favors Labor or the Conservatives. This does not, however, in general remove the desirability of randomness in assignments to district races. A party should not generally assign all its weakest candidates to its safest seats. A rival party could sometimes exploit such a deterministic approach to this task when districts are not too safe.
Parliamentary elections also are not only about winning enough seats to become the majority party, since law or custom dictates that cabinet ministers must be members of Parliament and having particular individuals serve as ministers may be desirable. Therefore, at least to some degree, parties care about having particular individuals win their elections. Again, this weakens the case for complete randomization without eliminating mixed strategies as equilibria of the assignment game. Dual chess matches might be another application of our model. However, an institutional rule severely restricts strategic assignments. A widely accepted international rating system dictates ranked assignments for most orga- nized matches. Thus, sincere assignments are enforced. It is doubtful that players manipulate ratings by purposely losing games in less important tournaments or matches to the possible strategic benefit of their team. However, the analysis above implies that this approach may not result in the most competitive possible match.
A rule change in U.S. dual team golf matches may have been enacted to thwart strategic play. Match play (analogous to team tennis scoring) was once common in team competition. Over the years, most competitions have switched to using team strokes to determine the winning team (the NCAA adopted this rule for post-sea- son matches in 1939).
Restrictions on assignments in tennis matches are minimal. Coaches in the NCAA can file grievances for facing an insincere line-up, which can lead to forfeitures. A coach can, however, defend a line-up based on performance in practice, disciplinary considerations, etc. Indirect limits on regular-season strategic play come from rules governing post-season tournaments. For the NCAA invita- tional team tournament, coaches submit player rankings along with their season records at each previously assigned position. Coaches have the opportunity to defend their player rankings in a meeting. Once a ranking is approved, the players must be so assigned throughout the tournament (there are more players than will play in each match). Post-season conference tournaments have similar rules. For
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example, to ensure that a player can be ranked number one in a post-season tournament, a coach needs to assign the player to the number-one court suffi- ciently often during the regular season. Top players are also ranked by state associations and some nationally. This places increased pressure on teams with these ranked players to assign players more sincerely.
For these reasons, teams with many strong players or with a serious chance of a post-season team title will use strategic assignments less often. However, our model still applies to teams with no hope of post-season invitations. When bad teams play good ones, the bad team then has an advantage, it faces an opponent whose assignment strategy is constrained while being able to choose any assign- ment it wishes. Consider, for example, a good team that is constrained in assigning its top m < n players, versus a weaker team facing no assignment constraints. Equilibrium has the weaker team assign nourandomly m judiciously selected players to the top m slots and both teams randomize with equal probabilities over the remaining n - m players and slot assignments. This asymmetry increases the difficulty for a strong team of compiling a very good record in dual matches over a season. This asymmetry, that bad teams are more able to use strategic assign- ments, may be one source of the common belief that only weaker teams resort to 'stacking.'
5. Conclusions
In our basic model, we find that the condition of pure strategy equivalence (no matter what pure strategy one player plays, the other player can attain every possible outcome) leads to the existence of a completely mixed strategy equilib- rium in which every pure strategy is played with equal probability. Furthermore, this equilibrium is generically unique. In team assignment games, pure strategy equivalence holds when winning each pairwise contest is equally valuable and the probability that one side wins a particular contest depends only on the identity of the two individuals in that contest. In both team sports and politics, there are many reasons to suspect that equal probabilities mixing does not occur. We have shown that modifying the strong assumptions of the simple model moves equilibrium strategies away from uniform mixing. However, when these other factors (un- equally valued seats, for example) are not too important, equilibria continue to be in mixed strategies.
In Section 1, we stated that, in races with incumbents, political parties would be playing the assignment game for the open seats. But this introduces an asymmetry between the parties. In Tennessee in 1994, there were simultaneous races for two U.S. Senate seats with one Democrat incumbent. The Republicans had two candidates for the two races and could effectively choose who faced the incum- bent. Thus, the Republicans had greater strategic flexibility, as does a tennis team of unranked players facing an opponent with ranked players.
J. Hamilton, R.E. Romano / European Journal of Political Economy 14 (1998) 101-114 113
Our model also sheds light on legislative elections for multiple districts when candidates do not face any residency requirements. Parliamentary elections in Australia, Canada and the United Kingdom take place under such rules (although tradition effectively requires candidates to reside in the same Australian state or Canadian province, each comprised of many districts). The tradition of resigna- tions by back-benchers to allow ministerial candidates in their party who lost their own races to compete for the seat in a by-election indicates the unimportance of residency within the district.
Each district election is a distinct contest, no matter whether first-past-the-post or another method is used to determine the winner. If each party's candidates' probabilities of winning an election against the potential rivals in the other party do not differ across districts, then our model would apply directly to two-party competition. However, there are clearly safe and marginal seats for each party: a given Labor candidate will have a much better chance against a given Conserva- tive candidate in a Labor stronghold than in a Conservative one. For small deviations from pure strategy equivalence, the unique Nash equilibrium still uses mixed strategies. Thus, we should expect to see a degree of randomizing behavior in parties' assignments of candidates to districts for general elections to Parlia- ment.
Our basic assignment game also provides a starting point to think about some economic problems. Duopoly assignment of scientists to projects in multiple patent races approximately fits the model if priors on the value of winning any patent race are diffuse and little discounting is present. The objective in assigning scientists to projects is then to maximize the number of races won. Assignment of lawyers to cases fits if the outcome of cases depends primarily on the match-up of lawyers and the objective is to maximize the number of cases won (as would occur if there are no contingency fees and reputation building is paramount). Assignment by two networks of shows to time slots would conform to the model if networks compete for shares of a viewership that does not vary over the time slots or with program pairings. The TV-scheduling problem suggests extending the problem to more than two networks. Suppose TV viewership and network shares depend on the vector of shows simultaneously broadcast, but are independent of the time slot. This might hold approximately over subsets of time slots. The implied N-network scheduling game (within the subset of time slots) need not be constant sum, but satisfies a generalization of pure strategy equivalence. Every feasible assignment of one network confronts the other networks with the equivalent game. Equal probability assignments are also an equilibrium in this game. However, other equilibria will generally exist. Characterizing equilibrium sets in this game is a topic for further study.
Admittedly, any of these problems deviates from our basic model, but the analysis provides some insight. For example, payoffs from winning particular patent races, legal cases, or time slots are likely to vary, but we have seen that this alone need not eliminate randomization. More generally, our model suggests the
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presence of a significant force for randomization in games entailing multiple contests.
Acknowledgements
We thank Peter Faynzilberg, Amihai Glazer, Alvin Roth, Jozsef Sakovics, Patrick Sileo, Michael Veall, participants in workshops at Carnegie Mellon University and Pennsylvania State University and two anonymous referees for valuable comments. We also thank coaches D. Blair, I. Duvenhuge, S. Perelman, P. Moss and R. Scheines for providing information about tennis and the NCAA and Kenneth Rogoff, grandmaster, for serving as our chess consultant. The first author thanks DGICYT (Spain) for financial support and Duke University and the Institute d'An~lisi Econbmica for their hospitality during this research. The second author thanks Carnegie Mellon University for financial support and hospitality during this research.
References
Bohnenblust, H.F., Karlin, S., Shapley, L.S., 1950. Solutions of discrete two-person games. In: Kuhn, H.W., Tucker, A.W. (Eds.), Contributions To The Theory Of Games. Princeton University Press, Princeton, NJ, pp. 51-72.
Friedman, J., 1990. Game Theory with Applications to Economics. Oxford University Press, New York.
![4.5. Contests [extras]](https://static.documents.pub/doc/80x56/55c4b0a3bb61eb182c8b45da/45-contests-extras.jpg)
