Chapters 29, 30 Game Theory A good time to talk about game theory since we have actually seen some types of equilibria last time. Game theory is concerned with the general analysis of strategic interaction. It can be used to study parlor games, political negotiation, and economic behaviors.

Normal form games: Prisoner’s dilemma. Defect 招 Coop 不招 Defect 招 -5,-5-1,-10 Coop 不招 -10,-1-2,-2

This game has a strictly dominant strategy. A dominant strategy is one which is optimal no matter what the other player does. Once a player has a st. dominant strategy, the task of predicting what other will do (the most interesting thing in game) becomes unnecessary. We may not be this lucky to have a st. dominant strategy all the time. But we might have a st. dominated strategy. So at least we can say we won’t play st. dominated strategies.

LR U2, 46, 3 M3, 33, 4 D5, 44, 3

One way to “solve” the game above is to use iterated elimination of strictly dominated strategies. So our story may go like, 1 is rational, so 1 will not play M. Given everyone knows this, 2 is rational, 2 will not play R. Given everyone knows this, 1 is rational, 1 will not play U. So (D, L) becomes our prediction. This process uses some common knowledge of rationality.

But, are you really a person who will play according to this logic of iterated elimination of strictly dominated strategies? Will you get the highest payoff if you do? Talk a bit about the games we played before and the cognitive hierarchy theory. Despite of this, theorists still feel that iterated elimination does not give sharp predictions.

BalletFootball Ballet2, 10, 0 Football0, 01,2

In battle of sexes game, iterated elimination does not help. A stronger solution, called Nash equilibrium, is often used. The concept of Nash equilibrium is this. A strategy profile (so every player has a part) is a Nash equilibrium, if given your opponent plays his equilibrium strategy, you play yours is a best response to that. In other words, players are best responding to each other.

Equivalently, we can say that given your opponent is playing equilibrium, you have no profitable deviation. That is exactly why we are looking at the reaction function in Cournot competition. A reaction function is simply a function that gives you the best response to every possibly plays of your opponent. We then intersect the two reaction functions to get Nash equilibrium where both are best responding.

In the battle of sexes game, there are two pure Nash equilibria, (ballet, ballet) and (football, football). Notice that in Nash equilibrium, a player’s is checking that he is best responding to his opponent’s play. In other words, implicitly we are assuming that a player’s belief is correct, or consistent with opponent’s play. So Nash equilibrium has two important components: best responding and correct beliefs.

There are some interesting empirical works supporting Nash equilibrium. For instance, football players in penalty kicks. Yet there are also reasons sometimes people don’t feel comfortable playing Nash equilibrium. Or sometimes we can even coordinate better than Nash. Look at the coordination games we played.

leftright top1, 01, 1 bottom-1000, 02, 1

In the literature, there is a well- recognized “problem” with Nash equilibrium about empty threats. Suppose in the first class meeting I came in and announced: “I want total dedication from my students, so you should drop all the other courses and just take my course. This will guarantee you spend all the efforts in my class. If you do not do so, I will not let you take the final and you will fail.”

Now it is a sequential game. You first decide whether you will drop other courses and then if you do not, I decide whether I will bar you from the final. My threat of not letting you take the final seems not credible (an empty threat) since at the final, what is the point of baring you from exam if my purpose is to increase your efforts in the course and efforts are done?

Moreover, NTU is a serious institution. So you expect that at the final, I will not carry out the threats and hence you will not drop other courses. That is a NE.

Now another Nash equilibrium looks like this. You take my threats seriously and you drop all other courses. Since you drop all courses I don’t need to carry out my threats of barring you from the exam. So my empty threat is never tested. I am very happy with the outcome. And you, given your pessimistic belief, are also doing what is the best for yourself. This does not seem very reasonable. So we want to rule out NE of this kind.

This brings us to Subgame Perfect Nash equilibrium (SPNE). A subgame is just like a game, so let us talk a bit about game trees. Drop ---- (2,5) Don’t Drop -----Bar (0,-5) -----Don’t Bar (4,2) So for SPNE, we solve backwards.

Talk about the sheep-lion story. The Stackelberg equilibrium is an SPNE. We are now equipped with NE and SPNE, it seems enough so what else can theorists do? There are still lots of things. For one, look at finite repetition of PD game. Defect all the time is the unique SPNE. But do you really believe in that?

One potential answer is to appeal to infinite horizon. Remember last time when we talked about repeated Cournot. With repeated interactions, as long as the future is important enough, we might be afraid of future punishment, which makes current cooperation possible. This is the result of Folk theorem. Another interesting direction is, do we really believe that people are so rational?

Can not-so-rational players play out equilibrium? Talk a bit about traffic and social conventions.