Maximin vs Nash: revisited

[Mehmet Mars Seven]

Dear all,

I’m pleased to share the following work with Shaun Hargreaves Heap, contributing to the historical discussion on “maximin vs. Nash.”

The strongest support for maximin over Nash equilibrium probably came from Harsanyi, who argued that he found maximin more rational in “unprofitable games,” in which the Nash equilibrium payoff is equal to the maximin guarantee, as well as from Aumann and Maschler.

Here we revisit the maximin vs. Nash discussion by focusing explicitly on actual payoffs rather than guarantees alone, and we ask a broader question: under what conditions does a maximin profile strictly outperforms all Nash equilibria? Our illustrative example is the following symmetric 3x3 game:

7,7  9,4  0,8
4,9  8,8  7,4
8,0  4,7  3,3

This game has a unique mixed Nash equilibrium strategy (1/2, 1/4, 1/4); and a unique maximin strategy (0, 5/8, 3/8).

The maximin profile gives each player a strictly higher payoff than the Nash equilibrium (6.125 > 5.75). Moreover, the maximin strategy is a best response to the Nash equilibrium strategy, in which case the Nash player receives strictly less than the maximin player does.

We believe that the maximin strategy is very attractive in this game so prefer it over the Nash strategy. We also show that this illustration is not atypical or idiosyncratic because, in our main result, the class of such games in which a maximin profile strictly Pareto dominates all Nash equilibria has the same cardinality as the class of games in which a Nash equilibrium strictly Pareto dominates all maximin profiles. We also provide several results on when maximin profiles and Nash equilibria coincide in terms of both payoffs and strategies.

A simple observation we emphasize is the following. The standard justification for Nash equilibrium over maximin is that it gives each player at least the player’s own maximin guarantee. However, maximin is even more powerful by this criterion: when there is a multiplicity of equilibria, not every Nash strategy profile actually delivers this guarantee, whereas every maximin profile does.

This is the first draft, so we especially welcome your comments, suggestions, and criticisms. Many thanks in advance.

Best wishes,
Mehmet Mars

[Karl Schlag]

hi mehmet
i must be missing sth as your strategy (0, 5/8, 3/8) gives 5,5. 
the maximin payoff cannot be strictly larger than that of some NE.
greetings, karl

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Karl Schlag  
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[Mehmet Mars Seven]

Hi Karl,

You're right that the maximin value cannot exceed a Nash equilibrium payoff; however, the payoff at a maximin profile can be strictly higher.

In this game, the maximin value for each player is indeed 5.5. However, at the maximin profile [(0, 5/8, 3/8),(0, 5/8, 3/8)], each player receives 6.125. This is strictly greater than the Nash equilibrium payoff, which is 5.75.

I hope this clarifies our point.

Thanks,

Mehmet