Nash, Nash, Nash!

We've already seen a bit about Nash equilibria, but I think it would do anyone well to learn a little bit more about the concept and more importantly about game theory.

Game theory, itself, is an incredible discipline. Like any science, it is best done by describing the world with math - something which must be a part of the heroic in man. There is nothing greater than seeing certain aspects of the world being represented in mathematical generalizations! To go from particular observations and instances to equations which explain the general, underlying principles of reality is very truly astounding. Game theory seeks to model one specific bit of reality: strategic decisions people must make, given the decisions of other people.

Let's look at a simple game. Every game has certain elements: a set of players, a set of strategies for each player, the information available to each player throughout the game, and payoff functions which show the players' preferences over outcomes. Of course, this information can be represented in several ways. Two common ways are known as the 'extensive form' and the 'normal form.' We'll look at the normal form here. Now, consider the following game.

We have two players, player 1 has strategies (X, Y) and player 2 has (A, B). This nifty bimtarix also shows their preferences over outcomes, with player 1's utility listed first.

The Stag Hunt

This is the normal form in a bimatrix. A few implications come with formalizing a game like this. The most important is that the players are making their decisions simultaneously, and in this case each player knows that this is the game, that the other player knows this, etc. The numbers represent payoffs, and the higher the better. Now! What would you say is player 1's best strategy? If you're thinking that the thought of a big, juicy nine means that player 1 should choose X, then that's good reasoning. Of course, we also must consider what player 2 might do. It's explicitly obvious that player 2 would also be enticed by the nine, and so player 1 would do well in choosing X. In fact, this is the Nash equilibrium and is what perfectly rational players would do. Interestingly enough however, people actually choose to perform either Y or B in reality. This really isn't economically rational, but the reasoning is that people aren't really 'Econs' (perfectly rational agents). Thus, other people may not see that X and A are best, and so people will then guarantee themselves a good ol' eight with Y or B. Of course, this outcome is inefficient. Thus, we have a disparity between economic rationality and our heuristic human sense of rationality. Fascinating!

A quick point: you should check to see if the strategy profile (X, A) really is the Nash equilibrium. Ask yourself, is X the best strategy, given that player 2 plays A? And then ask the inverse from player 2's point of view. You should conclude that, yes, X and A are the best responses, given the other player's strategy. I mention this because a crucial insight about the NE is that such equilibria are composed of best responses. This is where the concept gets its normative force (or its cutting power.) Think about it. I had to.

Anyway, that game is actually known as the 'stag hunt game', and is actually one of many famous little games meant to model common social situations. X and A are usually stipulated as 'Hunt stag' and Y and B are 'Hunt hare.' Here are a few more common games that you should get to know.

A Game of Douchebags

This is the game of chicken. Everybody has seen this in some lame movie. Two jerks race toward each other, trying to use their bravado to make the other person swerve. Here, X and A are 'stay the course', Y and B are 'swerve.' Clearly, the profile (X, A) means death for both and thus each get a payoff of 0. If one swerves and the other stays then one will get a payoff of three while the other is glad to be alive but embarrassed. and gets a payoff of 1. Should both swerve then the embarrassment won't be as strong and thus each gets two. This game has more worth than a mere description of two jerk-faces showing off. It can also be used in evolutionary game theory (well, most games can be), in which the players don't make decisions but are instead strategies which are pitted against each other. X and A would then be 'Hawk' and Y and B would be 'Dove'. If two Hawks meet, then they both lose due to a horrid fight. If a Hawk and Dove meet, then the Dove loses out but stays alive, and the Hawk gets some territory or something. If two Doves meet, then all is well.

Married... with Decisions!

Here we have a battle of the sexes. A wife and husband want to spend time with each other at an evening event, but a lack of cell phone service means they'll have to guess which event the other is going to. X and A are 'football' and Y and B are 'ballet.' Obviously, player 1 is then the man.

The Prisoners' Dilemma

We've already seen the Prisoners' Dilemma, but there's more to the game than my last exposition gave. The NE solution concept has already been spoken of, but there's another concept here which can be used. Notice how Y is always better than X. If a strategy is always better than all other strategies, then that strategy strictly dominates. If a strategy is sometimes better and never worse than other strategies, then that strategy weakly dominates. In this case, Y and B are both strictly dominating strategies, meaning that they are always better than the other strategy. In this way, Y and B are always best responses, and as we know, Nash equilibria are composed of best responses. Hopefully that gave you an "Aha!" moment, like it did to me. If not, I may just be special

The Prisoners' Dilemma actually provides a significant justification for having a state. The dilemma actually nearly models the situation of public goods (the zeros inaccurately represent payoffs here), things which make everyone in society better off. Think of clean air, roads, tornado warning systems, and even justice. We can think of the players in the game as the individual and everyone else. Let's look at the case of clean air. The individual and everyone can cooperate and ride their bikes to work. This leads to some pretty clean air. However, the individual sees that the air would still be nice 'n' fresh even if they chose to drive their Hummer powered by sliced dolphin to work, and so they may choose to 'Defect' (Y and B) instead of 'Cooperate' (X and A.) Of course, everyone else may adopt this reasoning also, and thus everyone will choose to defect. This leads to the suboptimal outcome of (Y, B), which is air that burns your scalp off. Of course, if a state exists, then it may impose some punishments if people choose this outcome. Marvel!

From the Prisoners' Dilemma we can glean another illuminating inference. For the most part, everyone chooses 'Cooperate.' The social norm is to cooperate, in fact. However, in the case of the Prisoners' Dilemma, this is oft because the state adds costs to defecting, so cooperating becomes preferable (and the Nash equilibrium.) Generally, social norms can be explained in terms of Nash equilibria. Take driving. When I go out to drive, I am confronted with the choice of driving on the left or right side of the street. Now, I've passed the rigorous driving test of the Arizona DMV, so I can tell you a bit about the issue. Driving on the right side of the street is my best response, given that everyone else does so also. The same can be said for all other drivers. Thus, the norm has been explained by the Nash equilibrium concept. Societies settle on Nash equilibria in various situations, including moral ones.

Now you have bared witness to the awesome power of game theory. We have considered cooperation, jerkface bravado, and an argument for the state all by looking at some small, unimposing bimatrices. If you're interested in learning more, then I wholeheartedly recommend this book. We'll come back to game theory from time to time, but this will suffice for now!