The Social Will-Testing Game and its Solution

We examine a two-person game we call Will-Testing in which the strategy space for both players is a real number. It has no equilibrium. When an infinitely large set of players plays this in all possible pairings, there is an equilibrium for the distribution of strategies which requires all players to use different strategies. We conjecture this solution could underlie some phenomena observed in animals. 1 The Will-Testing Game This game is played by two people in continuous time through a fixed time interval. Either player may capitulate at any time, thereby ending the play. The payoff for the non-capitulator is some constant times the amount of time remaining in the game; the capitulator gets half as much as the non-capitulator. The tension in the game is that both players want someone to capitulate in a hurry, but both want the other one to do so, and hence they both stall. There is no equilibrium to this game. The maximum sum of payoffs is 1 + .5, so players could judge their earnings by how much of that they obtained; many people would judge 0.75 to be a satisfactory take-away pay. In the case where this game is played repeatedly with the same player, there is an opportunity to split the 1.5 quite evenly in half, but achieving this requires the development of trust. How humans or chickens or other animals play this game is an area open to behavioral studies. The tension changes as we alter the fraction that the capitulator gets. Clearly if his fraction is 1, then it does not matter who capitulates first, and both will do so immediately. If his fraction very small (say 1/100) then there 1 ar X iv :1 20 6. 61 48 v1 [ cs .G T ] 2 7 Ju n 20 12 is a huge incentive to stall. If his fraction is close to 1 then it does not matter very much who goes first, and it’s relatively easy to make the sacrifice to go first. Since the fraction makes a big difference, we will parameterize the game and call the capitulator’s fraction ρ where 0 ≤ ρ ≤ 1. In spite of its description as being played in continuous time, we will formalize it as a one-shot game where each player i selects a real number si in the fixed interval [0, T ]. Then the payoffs for players 1 and 2 respectively are:  (T − s2) and ρ× (T − s2) if s1 > s2 ρ× (T − s1) and ρ× (T − s1) if s1 = s2 ρ× (T − s1) and (T − s1) if s1 < s2 The social version of the game has a large set of players who each choose a strategy once; their payoffs are the average of the payoffs using that strategy against all other players. This captures the idea of a group of social animals that play the game repeatedly with partners picked uniformly at random from the same group. There is no fixed pure equilibrium for this game when there are a finite number of players. In the limit of infinite players there is a density function for the equilibria, but it does not specify which player uses which strategy so all permutations are equivalent. Finding this distribution involves an equation that makes the usual assumption that all players get equal payoffs, so the players are quite indifferent to which permutation occurs. One strong interesting feature of the equilibria is that all players must choose different strategies. Birds and mammals and many other animals are generally perceived as being individuals who are all slightly different, populating a spectrum of behavior space. They are also widely seen as having social structures like dominance hierarchies wherein pairwise relations between individuals can be characterized. We view the social version of Will-Testing to be an example of a game that produces a similar ordered structure where the existence of differences in individual behaviors is a straightforward requirement of the game’s solution. In the case of this particular game, the social dynamic will be interpretable as a total ordering (like a pecking order) simply because the strategy space is real and one dimensional. 2 Deriving the Equilibria We seek a density over the strategy space, den(t) ≥ 0 and ∫ T 0 den(t)dt = 1 . To find the average over all possible pairwise interactions in the social version of the game, we must integrate over that space. Define the pay of strategy s