NEJAT ANBARCI
DIVIDE-THE-DOLLAR GAME REVISITED ABSTRACT. In the Divide-the-Dollar (DD) game, two players simultaneously make demands to divide a dollar. Each player receives his demand if the sum of the demands does not exceed one, a payoff of zero otherwise. Note that, in the latter case, both parties are punished severely. A major setback of DD is that each division of the dollar is a Nash equilibrium outcome. Observe that, when the sum of the two demands x and y exceeds one, it is as if Player 1’s demand x (or his offer (1−x) to Player 2) suggests that Player 2 agrees to λx < 1 times his demand y so that Player 1’s demand and Player 2’s modified demand add up to exactly one; similarly, Player 2’s demand y (or his offer (1−y) to Player 1) suggests that Player 1 agrees to λy x so that λy x+y = 1. Considering this fact, we change DD’s payoff assignment rule when the sum of the demands exceeds one; here in this case, each player’s payoff becomes his demand times his λ; i.e., each player has to make the sacrifice that he asks his opponent to make. We show that this modified version of DD has an iterated strict dominant strategy equilibrium in which each player makes the egalitarian demand 1/2. We also provide a natural N-person generalization of this procedure. KEY WORDS: Divide the dollar, Nash equilibrium, The egalitarian outcome, Iterated dominance equilibrium
1. INTRODUCTION
In the Divide-the-Dollar (DD) game (which is a simple version of ‘Nash’s (1953) Demand Game’) two players, 1 and 2, simultaneously make demands in order to divide a dollar. If the sum of the demands does not exceed one, each player receives his demand; otherwise, each player receives a payoff of zero. DD is a prototype of a very general problem: how should the gains from cooperation be divided among the participants. As Binmore (1998) puts it: “Bargaining between two individuals is worthwhile when an agreement between them can create a surplus that would otherwise be unavailable. ... The archetypal version of this problem is called dividing the dollar.” (p. 21.) “Many bargaining problems have this simple structure. For example, wage negotiations often reduce to a dispute Theory and Decision 50: 295–304, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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over how the surplus created by the joint efforts of a firm and its workers should be divided.” (p. 69.) This game, however, has a multiplicity of Nash equilibria since every division of the dollar is a Nash equilibrium outcome (as a matter of fact, (0,0) is also a Nash equilibrium outcome; in that equilibrium, each player demands the entire dollar). There has been several attempts to rectify that problem.1 Most notably, Brams and Taylor (1994), in this Journal, proposed three modifications of DD all of which yield the egalitarian outcome (but not necessarily egalitarian demands) under different equilibrium concepts.2 In addition, Brams and Taylor (1994) have the remarkable result that (if iterated weak dominance – and thus iterated strong dominance – is used as the equilibrium concept) there is no scheme that induces egalitarian demands which also satisfies the following conditions: 1) Equal treatment of equal demands, 2) No player’s payoff is more than his demand, 3) If the sum of the demands does not exceed one, then each player receives his demand, 4) If the sum of the demands exceeds one, then the dollar, nevertheless, is completely disbursed to the players. 5) If all demands are greater than the egalitarian level demand, the player with the highest demand does not receive a higher payoff than the player with the lowest demand. Two of Brams and Taylor’s mechanisms, DD1 and DD3, satisfy Conditions 1–5, and as their result above implies, fail to induce egalitarian demands under iterated dominance equilibrium. Their remaining mechanism, DD2, satisfies Conditions 1–3, and 5 as well as the following condition (instead of Condition 4): 4 ) If the sum of the demands exceeds one, then each player receives less than his demand.3 DD2 induces egalitarian demands if either player demands the egalitarian level demand in the initial stage. However, some subgames of that game do not have a pure-strategy Nash equilibrium and DD2 does not necessarily induce egalitarian level demands when the number of players N is greater than 2. In this paper, we analyze another mechanism which (i) satisfies Conditions 1–3, and 4 (like DD2),4 and (ii) induces egalitarian
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demands under iterated dominance equilibrium even when N > 2 (unlike DD2). Our mechanism changes the payoff assignment rule of DD. In DD, observe that, when the sum of demands exceeds one, it is as if Player 1’s demand x (or his offer (1−x) to Player 2) suggests that Player 2 agrees to λx < 1 times his demand y so that Player 1’s demand x and Player 2’s modified demand λx y add up to exactly one (in a sense, λx represents what proportion of Player 2’s demand Player 1 considers as reasonable); similarly, Player 2’s demand y (or his offer (1−y) to Player 1) suggests that Player 1 agrees to λy x so that λy x+y = 1. Our new rule is such that, in the case when the sum of demands exceeds one, Player 1 receives λx x and 2 receives λy y (i.e., each player has to make the sacrifice that he asks his opponent to make); when the sum of demands does not exceed one, each player receives his demand, as in the classic DD game. We show that this scheme has an iterated strict dominant strategy equilibrium in which each player makes the egalitarian level demand 1/2. We also provide a natural N-person generalization of this scheme.
2. A MODIFICATION OF DD
Although our results hold for the N-person case, where N2 (see the Appendix), here we will only consider the intuitive 2-person case. Thus, consider the following one-stage modification, DD∗ , of DD: DD∗ : Two players 1 and 2 simultaneously announce their demands x and y, such that x,y∈(0,1].5 If x+y 1, each player receives his demand. Let λx be the largest non-negative number such that λx y+x = 1 (i.e., λx = (1−x)/y); similarly, let λy be the largest nonnegative number such that y +λy x = 1. If x+y > 1, Player 1 receives a payoff π (x) = λx x and Player 2 receives a payoff π (y) = λy y. When x+y > 1, λx represents how much Player 1’s offer (1−x) to Player 2 complies with Player 2’s demand y. Then in turn Player 1 receives the payoff λx x. Observe that a player definitely receives more than a zero payoff unless he demands the entire dollar (recall that demanding a zero payoff is ruled out). Compare this to the original DD game where given x+y > 1 each player gets a zero payoff regardless of by how little the sum of the demands exceeds
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one. Lemma 1 states that when the sum of the demands exceeds one, the resulting payoff combination is feasible but not efficient: LEMMA 2.1. Suppose x+y > 1. Then λx x + eλy y < 1. Proof. 6 Without loss of generality, assume that λx λy . Then x + λy y x + λx y.
(1)
By definition of λx , x+λx y = 1, and by assumption, x + y > 1; thus, we must have λx < 1. Hence, λx x + λy y < x + λy y.
(2)
But then, using (2) and (1), and the fact that x+λx y = 1, we get λx x + λy y < x + λy y x + λx y = 1. We want to show that in DD∗ ’s iterated strict dominant strategy equilibrium7 each player’s demand is 1/2. The following is our main result in the two-player case (a natural generalization to any N > 2 can be found in the Appendix): THEOREM 2.2. DD∗ ’s iterated strict dominance equilibrium is (1/2,1/2).8 Proof. 9 Without loss of generality, consider Player 1’s decisionmaking. Observe that x(1−x) achieves its maximum at x∗ = 1/2. Given that fact, our proof follows from the following two claims: Claim 1: A demand of x 1/2 and x+y < 1: Because y > 1/2, y(1−y) < 1/4, so 1−y < 1/4y. Because x+y < 1, π (x) = x < 1−y. Therefore, π (x) < 1−y < 1/4 y = π (x∗ ). Case 2b. y > 1/2 and x+y > 1: Then π (x∗ ) = 1/4y > x(1−x)/y = π (x) since x(1−x) is maximized at x∗ = 1/2. This completes the proof that a demand of x < 1/2 is dominated by a demand of 1/2 for Player 1. Claim 2: Once the dominated strategies x < 1/2 and y < 1/2 are
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eliminated, a demand of x > 1/2 is dominated by the demand x∗ = 1/2. Since x+y > 1, Player 1 gets π (x∗ ) = 1/4y with a demand of x∗ = 1/2, and π (x) = x(1−x)/y with a demand of x. Again, the result follows from the fact that x(1−x) is maximized at x∗ = 1/2. Remark. Observe that when a ‘fair’ player demands 1/2 against a ‘greedy’ player who demands more than 1/2 (which of course is out-of-equilibrium play), the ‘fair’ player may get a lower payoff than the ‘greedy’ player (although demanding 2 is still the ‘fair’ player’s best response to the ‘greedy’ player’s demand). To see that suppose Players 1 and 2 demand 1/2 and 3/5 respectively. Then Player 1 receives (1/2)[(1/2)/(3/5)] ∼ = .41, and Player 2 receives (3/5)[(2/5)/(1/2)] ∼ = .48. Furthermore, if the ‘fair’ player’s opponent is even greedier, the ‘fair’ player’s payoff can be even lower than what he can achieve against a less greedy player. To see that suppose Player 1 still demands 1/2 but Player 2 demands 4/5. Then Player 1 receives (1/2)[(1/2)/(4/5)] ∼ = .31, and Player 2 receives ∼ (4/5)[(1/5)/(1/2)] = .32. 3. CONCLUSION
Brams and Taylor (1994) have the remarkable result that (if iterated weak dominance – and thus iterated strong dominance – is used as the equilibrium concept) there is no scheme that induces egalitarian demands which also satisfies Conditions 1–5 (as stated in the Introduction). DD∗ is a one-stage game where not only the equilibrium outcome but also the equilibrium demands are egalitarian; the equilibrium concept used, iterated strict dominance, only requires that the players successively eliminate their strictly dominated strategies. Our DD∗ scheme is unlike Nash’s DD in that, in case of disagreement, unless a player demands the entire dollar, DD∗ does not assign a zero payoff to any player. At the same time, DD∗ satisfies Condition 4 rather than Condition 4: i.e., it does not completely disburse the dollar to the players either. Preferring Condition 4’ over 4 implies that one does not treat two overbidding players (e.g., each demanding the entire dollar) more favorably than two underbid-
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ding players (e.g., each demanding very little), whereas Condition 4 treats the former-type players more favorably than the latter-type players. Also, not having Condition 5 implies that one may not necessarily favor any of the overbidding players. Although a case can be made for each set of conditions, one thing is clear: when Condition 4 is adopted instead of Conditions 4 and 5, reasonable modifications of DD that induce egalitarian demands under iterated dominance equilibrium can be constructed.10 Although DD∗ has a unique iterated dominance (and hence a unique Nash) equilibrium and satisfies desirable properties such as 1–3, and 4 , as the Remark illustrates, (in out-of-equilibrium play) DD∗ does not guarantee a higher payoff for the ‘fair’ player against a ‘greedy’ player (furthermore, if the ‘fair’ player’s opponent is even greedier, the ‘fair’ player’s payoff can be even lower). But a ‘fair’ player still does much better against a ‘greedy’ player in DD∗ than in the standard DD game, for in the latter the ‘fair’ player gets a zero payoff. On the other hand, a ‘fair’ player does even better against a ‘greedy’ player in DD1 and DD3 of Brams and Taylor than in DD∗ , but recall that DD1 and DD3 fail to induce egalitarian demands under iterated dominance equilibrium. 4. APPENDIX: AN N-PERSON GENERALIZATION
N-person DD∗ :11 N > 1 players simultaneously announce their demands xi such that each xi ∈ (0,1], where i = 1,..,N.12 If x1 +x2 +...+xN 1, each player receives his demand. Let αi be the largest positive number such that αi (x1 +x2 +...+ xi−1 +xi+1 +...+xN ) + xi = 1 (i.e., αi = (1−xi )/(x1 +x2 +...+ xi−1 +xi+1 +...+xN )). Let λi = (αi )N−1 . If x1 +x2 +...+xN > 1, each Player i receives π (xi ) = λi xi .13 LEMMA 4.1. Suppose x1 +x2 +...+xN > 1. Then λ1 x1 +λ2 x2 +...+ λN xN < 1. Proof. Rearrange all demands such that λi λi+1 (i.e., λ1 λ2 λ3 and so on). Then x1 + λ1 x2 + λ1 x3 + ... + λ1 xN x1 + λ2 x2 + λ3 x3 + ... + λN xN . By definition of αi , αi (x1 +x2 +...+ xi−1 +xi+1 +...+xN ) + xi = 1, and by assumption, x1 +x2 +...+xN > 1; thus, αi < 1. Since λi = (αi )N−1 ,
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we must have λi αi and in particular λ1 < 1. Hence, x1 +λ1 x2 +λ1 x3 + ... +λ1 xN > λ1 x1 +λ1 x2 +λ1 x3 + ... +λ1 xN . Then, using (2 ) and (1 ), and the fact x1 +α1 (x2 +...+ xN ) = 1 that implies x1 +λ1 (x2 +...+ xN ) 1, we get 1 x1 + λ1 x2 + λ1 x3 + ... + λ1 xN > λ1 x1 + λ1 x2 + λ1 x3 + ... + λ1 xN . THEOREM 4.2. N-person DD∗ ’s iterated strict dominance equilibrium is (1/N,...,1/N). Proof. Consider some Player i’s decision-making. Observe that xi (1−xi )N−1 achieves its maximum at x∗i = 1/N. Given that fact, our proof follows from the following two claims. Without loss of generality, assume that i = 1. Also, to simplify the notation, let x denote x1 and let y denote x2 +...+xN . Claim 1: A demand of x < 1/N is dominated by a demand of x∗ = 1/N for Player 1. Case 1. y 1−1/N: Clearly, Player 1 does better with a demand of x∗ = 1/N than a demand of x < 1/N. Case 2a. y > 1−1/N and x+y < 1: A demand of x∗ = 1/N yields a payoff of π (x∗ ) = (1/N)[(1−1/N)/y]N−1 and trivially a demand of x yields a payoff π (x) = x for Player 1. Since x+y < 1, we have y < 1−x. The latter implies 1/y > 1/(1−x); since both sides are not less than 1 and since N > 1, we also have (1/y)N−1 > [1/(1-x)]N−1 . Recall that x(1−x)N−1 achieves its maximum at x∗ = 1/N; thus, (1/N)(1−1/N)N−1 > x(1−x)N−1 when x is less than 1/N. Given that fact, multiply the left-hand side of (1/y)N−1 > [1/(1−x)]N−1 by (1/N)(1−1/N)N−1 and the left-hand side by x(1-x)N−1 to get (1/N)[(1−1/N)/y)N−1 > x[(1−x)/(1−x)]N−1 = x = π (x). Thus, π (x∗ ) > π (x) when y > 1−1/N and x+y < 1. Case 2b. y > 1−1/N and x+y > 1: Then π (x∗ ) = (1/N)[(1−1/N)/ y)N−1 > x[(1−x)/y]N−1 = π (x) since x(1−x)N−1 is maximized at x∗ = 1/N. This completes the proof that a demand of x < 1/N is dominated by the demand 1/N for Player 1. Claim 2: Once the dominated strategy xi < 1/N for each Player i is
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eliminated, a demand of x > 1/N is dominated by a demand of x∗ = 1/N for Player 1. Since x+y > 1, Player 1 gets π (x∗ ) = (1/N)[(1−1/N)/y)N−1 with a demand of 1/N and π (x) = x[(1−x)/y]N−1 with a demand of x. Again, the result follows from the fact that x(1−x)N−1 is maximized at x∗ = 1/N.
ACKNOWLEDGMENTS
I am grateful to a co-editor and two anonymous referees who made many substantial and fruitful suggestions. I am also thankful to John Boyd III, James Felli, Cem Karayalcin and Devashish Mitra for their useful comments and suggestions.
NOTES 1. The motivation for such mechanisms is as follows. ‘Dividing the dollar evenly’ corresponds to an outside agent, an arbitrator, who tries to help the parties to resolve their dispute via a mechanism that will induce the parties to reach an egalitarian solution on their own. (One referee suggested this justification.) In the industrial relations literature, for instance, providing an incentive for the parties to resolve their dispute voluntarily is considered an important virtue of an arbitration mechanism (see, for instance, Bloom (1981) and the references therein). In addition, ideally such a mechanism, unlike the original DD, should not be too punitive. 2. In DD1, the player with the lowest demand (or the group of players tied for the lowest demand) is paid off first if there is enough money to do so; if not, the dollar is divided evenly among the group. One then moves to the player with the second-lowest demand (or the group of players tied for the second-lowest demand) and proceeds in the same way with the money left from the player(s) with the lowest demand; and so on until the dollar is exhausted. DD2 scheme can be described as follows: Stage 1: Players 1 and 2 simultaneously make initial demands, x and y, which are made public. Stage 2: Players 1 and 2 simultaneously make final demands, a and b, but they are restricted in their choices to just x or y. If a + b 1, Player 1 receives a and Player 2 receives b; otherwise each player receives a zero payoff. One gets DD3, when DD2’s second stage is combined with the payoff assignment rule of DD1.
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3. The description of DD2 in the previous footnote reveals that it satisfies Condition 4 a la original DD: in case of disagreement, DD2 punishes both players severely regardless of how close the total of the demands are to one. 4. DD2, in addition to Conditions 1–3, and 4 , satisfies Condition 5. See the Conclusion section for a discussion that relates Conditions 4, 4 and 5. 5. x = 0 and y = 0 are ruled out so that each λ is well-defined. 6. One referee suggested this more accessible reorganization of Lemma 1’s original proof. 7. The iterated strict dominant strategy equilibrium of a game is always unique since the order of elimination of strictly dominated strategies is irrelevant. 8. To see why this strategy combination is not a dominant strategy observe that, when, say, Player 2 demands 1/3 it is not a best response for Player 1 to demand 1/2 (i.e., Player 1’s best response then is to demand 2/3). Our proof, however, will illustrate that (i) demanding less than 1/2 is a dominated strategy, and (ii) once any such demands are eliminated, demanding more than 1/2 becomes dominated. Thus, after the elimination of any demands less than 1/2, demanding 1/2 becomes the dominant strategy. (I am thankful to the editor for bringing up this issue of “how many steps it takes for demanding 1/2 to become the dominant strategy.”) 9. One referee and the co-editor suggested this direct proof instead of a long and cumbersome proof of this theorem in the original version of this paper. 10. One might consider a modification of DD∗ that satisfies Condition 4’: Suppose that the payoff assignment rule of DD∗ is modified such that if Player i’s λ exceeds the other player’s λ in the case of disagreement, Player i receives his demand x and his opponent receives (1−x) – in case of any ties between maximum λ’s, a coin flip can determine the winner. This scheme satisfies Condition 4 , and it is easy to see that the unique Nash equilibrium of this modified scheme is such that each player demands 2. Compared to DD∗ , however, this scheme has two shortcomings. First, demanding 2 is not an iterated strict dominance equilibrium strategy in this modified scheme. To see that consider the case N = 2 and suppose that Player 1 demands .8. Then Player 2’s best response is not demanding 2 (which would be the best response in DD∗ ); rather, Player 2’s best response now is to demand .79. Second shortcoming of this scheme is that it does not possess a continuous payoff function that DD∗ does. To see that consider the case where Player 1 demands .9: if Player 2 demands .91, the payoffs are (.9,.1); if Player 2 demands .89, the payoffs become (.11,.89) (note that all payoffs in these cases would be very close to each other in DD∗ ). 11. One referee suggested this N-person generalization. 12. xi = 0 is ruled out so that each λ is well-defined. 13. Note that λi = (αi )N−1 is required here because λi = αi alone is not sufficient to make x∗ = 1/N the iterated strict dominance equilibrium strategy in the N > 2 case. To see that consider the 3-person DD∗ with the demands (1/3, 2/3, 2/3). Notice that α1 = 1/2 α2 = α3 = 1/3. Without the exponent N-1 (i.e., with λ1 =
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α1 , λ2 = λ2 , λ3 = α3 ), observe that Player 1 would get a payoff of 1/6 (i.e., 1/3 times 1/2) and Players 2 and 3 each would get a payoff of 2/9 (i.e., 2/3 times 1/3). Then, demanding more than 1/3 (i.e., more than 1/N) would be more beneficial than demanding one’s fair share of 1/3. With the exponent N−1 (i.e., with λ1 = (α1 )2 = 1/4, λ2 = (α2 )2 = λ3 = (α3 )2 = 1/9), however, observe that Player 1 would get a payoff of 1/12 (i.e., 1/3 times 1/4) and Players 2 and 3 each would get a payoff of 2/27 (i.e., 2/3 times 1/9).
REFERENCES Binmore, K. (1998), Game Theory and the Social Contract II, The MIT Press, Cambridge, Mass. Bloom, D. (1981), Is Arbitration Really Compatible with Bargaining, Industrial Relations 20: 233–244. Brams, S. and Taylor, A. (1994), Divide the Dollar: Three Solutions and Extensions, Theory and Decision 37: 211–231. Nash, John (1953), Two-Person Cooperative Games, Econometrica 21: 128–140.
Address for correspondence: Nejat Anbarci, Department of Economics, Florida International University, Miami, FL 33199, USA Phone: (305) 348-2735; Fax: (305) 348-1524; E-mail:
[email protected] 