Theory and Decision

, Volume 50, Issue 4, pp 295–303

# Divide-the-Dollar Game Revisited

Article

## 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 λyx so that λyx+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.

Divide the dollar Nash equilibrium The egalitarian outcome Iterated dominance equilibrium

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### REFERENCES

1. Binmore, K. (1998), Game Theory and the Social Contract II, The MIT Press, Cambridge, Mass.Google Scholar
2. Bloom, D. (1981), Is Arbitration Really Compatible with Bargaining, Industrial Relations 20: 233-244.Google Scholar
3. Brams, S. and Taylor, A. (1994), Divide the Dollar: Three Solutions and Extensions, Theory and Decision 37: 211-231.
4. Nash, John (1953), Two-Person Cooperative Games, Econometrica 21: 128-140.