Wednesday 21 September 2011

Game Theory Naming Rights (geek post) UPDATED

We have added lecture time and content to our intermediate micro sequence this year, and as a result I find myself teaching basic game theory for the first time in 15 years. I have some old notes that introduces a number of canonical games using simple examples of 2x2 simultaneous games: these include a couple of versions of Pure Coordination, a zero-sum game (which I call Pure Conflict), Battle of the Sexes and, of course, The Prisoner’s Dilemma.

These are all standard textbook games. I also have three non-standard games. The first, which I call PureHarmony, has the following form:
 PH C1 C2 R1 (2,2) (2,1) R2 (1,2) (1,1)

It involves a dominant-strategy equilibrium in which either player’s choice of strategy has no impact on the other’s payoffs. This is a totally uninteresting game, except as a kind of dog that didn’t bark in the night: That is, it is a game with neither conflict nor a need for coordination, the two things that make game theory interesting.

The other two non-standard games are variants on the Prisoner’s Dilemma. I call the first of these Selfishness is ts Own Reward. I don’t have a name for the second, and so I am offering up naming-rights to anyone who can come up with a good story to motivate it. The Prisoner’s Dilemma, Selfishness is its Own Reward, and the final game have the following forms:

 PD C1 C2 R1 (2,2) (4,1) R2 (1,4) (3,3)
 SoR C1 C2 R1 (3,3) (4,1) R2 (1,4) (2,2)

 ? C1 C2 R1 (2,2) (4,1) R2 (1,3) (3,4)

Selfishness is its Own Reward has most of the attributes of a Prisoner’s Dilemma: a dominant strategy equilibrium in which the dominant strategy for each player imposes costs on the other player. But in SoR, unlike the PD, the selfish benefits to oneself arenot outweighed by the costs imposed by the other player, so that the equilibrium Pareto dominates the outcome in which both players behave non-selfishly. An example is a two-firm advertising game in which each firm’s advertising both takes market share from the other and brings new consumer’s into the market, with the benefit of the new consumers outweighing the cost of the advertising. SoR is also useful as an illustration for how Kant's categorical imperative removes a sort of technical loophole from the Biblical golden rule. Taken literally, do unto others as you would have them do unto you  would imply both players playing Strategy 2 in Selfishness is its own Reward, but the categorical imperative would not.

The final game has the same outcome as a Prisoner’s Dilemma—the unique equilibrium is Pareto dominated by one in which the players behave non-selfishly, but the equilibrium is arrived at by iterated elimination of dominant strategies, rather than both players having a dominant strategy. In a separated Prisoner’s context, Column would not benefit from finking on row if he thought Row would not fink, but knowing that finking is a dominant strategy for Row, finking is still the optimal strategy for Column.

So I have two questions for the game theory geeks among you. First, has anyone seen either Pure Harmony or Selfishness is its Own Reward before, and if so what have those games been called. And second, can anyone think of a good economic example that has the structure of the final game, and if so, what should it be called?

UPDATE:  A correspondent who could not access the comments has tweeted to suggest that the final game has the form of the Stag Hunt. It is close to a stag hunt, but not exactly. The Stag Hunt is what I have called Pure Coordination II in my notes—a game in which there are multiple, but Pareto rankable, Nash equilibria. The unnamed game above has a Pareto dominated Nash equilibrium but it is a unique equilibrium without the Prisoner’s Dilemma attributed of dominant strategies for both players.