Game Theoretic Modeling of Social Capital

As someone who has lived in three major U.S. cities and one European city, I can tell you that there is wide variation in the societal fabric that holds large metropolitan areas together, or, broadly speaking, social capital.  Going along those stereotypes, San Franciscans are generally nice to everyone because who knows if you are talking to the next billionaire-tech-start-up-whiz; Chicagoans are polite but maybe a bit distant, because – I’m not sure exactly, but it’s what you do in Chicago; New Yorkers are pushy and aggressive, but, I’ve been told, come together like nobody else in times of crisis; and Parisians are polite until, well, they figure out that they are addressing a non-Parisian.  You get the idea.

So what determines these day-to-day interactions?  Consider the famous prisoner’s dilemma model, as seen in Figure 1.  For simplicity consider two people, who face symmetric choices: they can either cooperate or defect.  If they both cooperate, they both get a payoff of 5 (these payoffs are arbitrarily assigned for illustration purposes); if one cooperates and one defects, then the defector gets a payoff of 6 and the cooperator gets a payoff of 0; and if they both defect, they both get a payoff of 1.  In a one-shot game, it is clear that the expected payoff from defecting is higher, so both players opt to defect and both get a payoff of 1.

game-theory-pd.png

From a social capital perspective, it makes more sense to think about this as an infinitely repeated game, since members of a given society are constantly interacting.  Let’s say in a social capital sense that cooperation is being nice and defection is being mean.  You could imagine a scenario where people are nice until someone is mean to them, at which point they are jaded and are mean to everyone forever after (i.e. the Grim Trigger strategy in Game Theory).  Unless you are a saint, this is probably an accurate depiction of human behavior.  In an infinitely repeated game, players have a higher incentive of being nice – the payoff difference between 5 and 1 gets compounded with each time period, just like it is much more pleasant to go through your day when everyone is nice than when everyone is mean.

So what determines whether the incentive is high enough for a given person to be nice, rather than mean?  The answer lies in how much each player values the future. In other words, every player values today more than tomorrow or the day after, but the discount of present values varies.  Let’s call this discount factor δ (delta): if δ is high, then the person only cares about the present, and if δ is low, then the person greatly cares about the future.  Therefore, the discounted present value for being mean is expressed by 1/(1-δ) and the discounted present value for being nice is expressed by 5/(1-δ).  If δ is high enough, then deviation to being mean is more profitable than continuously being nice.   Based on these expressions, in this case, the magical δ cutoff value is 1/5; if a person’s δ is greater than or equal to 1/5, then they will be mean, otherwise they will be nice.

There are various ways to “measure” someone’s valuation of the future, or δ.  For example, one could follow Harvard sociologist Robert Putnam’s lead from Bowling Alone and do bowling league and other association memberships.  Another idea could be the length of time persons expect to stay in the neighborhood.  But, at the end of the day, each individual within each social grouping with its own social capital rules has their own “true” (but empirically unknown) value of δ.  Model the δ, and you could model the social capital of a society.

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