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Odd Man Out

In Uncategorized on April 20, 2010 by Jenny

This may be a pretty naive question; so bear with me.

Yasha writes that the three-person Odd Man Out game is a better model than the Prisoner’s Dilemma or Stag Hunt for modeling coalitions (partly) because it can capture chimp behavior of coalition-forming.  But I wonder whether Odd Man Out can even do that.

In chimpanzee societies, the alpha male has the best reproductive success; so all the guy chimps want to be him.   To get an edge on the competition, some male chimps form agreements with other male chimps to team up and beat the rest out.  When one coalition is the only left standing, the two members duke it out.  The victor becomes the alpha male.  The other guy is beta.

But because Beta wants to be alpha, he forms a coalition with another male chimp, and together they overthrow Alpha.  Then, the other chimp becomes New Beta and Beta becomes New Alpha.  Unfortunately, then New Beta wants to be the alpha (so he can get girls), so he overthrows New Alpha with the help of another friend, and this process continues indefinitely (or until one chimp is beaten to death.  Gruesome.).

In the chimp scenario I’ve outlined, there are more than 3 players (Alpha, New Alpha, New Beta, & soon-to-be Newest Beta).  But Odd Man Out can only model three-player games (as far as I know).  So why would we think that Odd Man Out can explain such behavior?  Is it because only three players are being modeled at any given time?  But we could easily model four.  We could model soon-to-be-Newest Beta trying to convince New Beta to help New Alpha overthrow Alpha so New Beta can be the next alpha.  Why would we stop at three players?  Is the reason for stopping at three players any better than Skyrms’s and Binmore’s reasons for stopping at two?

Thoughts?

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4 Responses to “Odd Man Out”

  1. I have a similar question.
    The Odd-Man Out game seems to deal with a more general situation of three-person coalitions. The underlying incentive for gaining up is the sectioning of more resources per individual, but the presence of the third person leaves room for bargaining. However, the chimpanzee society is hierarchical, it has an additional structure that is not built into the Odd-man-out game. Yasha acknowledges the seemly inconsistency between the predictions of the odd-man-out game ( that there will be rapid coalition shifts ) and the chimp society (slow coalition shifts due to alpha male intervention), might it be a problem of this additional condition that is not modeled into the odd-man-out game?

    One response might be that since the odd-man-out game is a general three person game, the hierarchical society situation would be a possible outcome of the odd-man-out game. That is, the hierarchical structure ITSELF is a result of a three-person game and thus the origin can be explained by game theory. Combining this response with Jenny’s question, it is possible for the hierarchical structure to be explained by the odd-man-out game?

    • I now understand that the alpha male in the chimp society serves as the incentive for coalition formation, and the target of study is coalition formation under “a” incentive (not any particular one, including being alpha male or not). Thus the hierarchical dominance system has nothing to do with the question asked. Thus, Yasha’s solution to the problem of coalition shift frequency is showing how a property of this hierarchical dominance system (the incentive of the alpha male to maintain its status) is the reason why there is a disparity between prediction and actuality. In this case, the model does not fail to explain the coalition maintenance in chimp society after taking into account the contingent properties of the actual system itself.

  2. I think Yasha’s using a three-person game because there’s evidence of coalition formation and disintegration in chimp populations with as little as three members. You’re certainly right to think that most populations will have many more than the three members called for by Odd Man Out, but I think Yasha’s argument is that if chimps can form coalitions with only three individuals (c.f. the zoo example in Yasha’s paper), than such behavior will scale upwards. Introducing more chimps to the population will make coalitions more complex, and this may make studying coalitions more difficult, but I think the larger point will still stand.

  3. Just a couple of quick thoughts. First, Yasha’s conclusion isn’t necessarily (though maybe is implicitly) that three person games are the best models of coalitions. His paper will be an important contribution even if it merely establishes that the two-person models are clearly inadequate and three-person models certainly appear to meet the criteria (which are quite plausible) much better.

    Moreover, using a three person game seems to match the dynamics of single interactions very well. The options available and the pay-off structure are nicely captured – as you say, only three individuals are interacting and having thier fitnesses affected at any one time. That is, the decision at that time only directly affects the fitnesses of the three players currently playing (the fourth individual is only affected later on and by other decisions). Now certainly this game can be iterated over and over in order to capture the type of situation you describe. Yet, still only three players are involved in any given interaction. Although players could still play different strategies across iterations of the game, the payoff structure of single interactions is still nicely represented by the three-person structure.

    Finally, why not four or five or six person games? For one thing they would seem to be unnecessarily complex. If we want to explain the behavior with a game theoretic model simplicity is important (as Zac notes in his paper). Stripping away irrelevant complexities is thought to yield better explanations (unless the more complicated models provide more accurate predictions of the data). Furthermore, even if the most basic three person game is insufficient in some ways, we can supplement it isimilar to the ways that Zac supplements the basic model of altruistic behavior. By building in various biological observations we may get very good models of coalitions and given the basic structure of the fitness interactions involved in coalitions, the three person game seems like the best place to start.

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