IO Chapter 4

KEY TERMS:

Game theory - when two agents are trying to decide what action to take, as well as taking into account the possible actions of other agents. Many card or board games require players not only to think about the impact of their own move, but how the other player is likely to react. Sometimes actors have to decide at the same time, and live with whatever the outcome of their decision and the other player's decision is. Ex: paper/scissors/rock.

dominant strategy - the case where no matter what the other player does, the first player has a preferred strategy. You can determine this by looking only at the pay-off matrix of that player, to see if one strategy consistently pays off more than the other. Another possibility is to do an exercise where you ask what the first player's choice would be for each of the other player's options and see if there is a consistent pattern.

prisoner's dilemma - the situation where players end up in a suboptimal solution by pursuing their own best interest. Another way of thinking about is, is that the joint solution is different than the individual solution. Or as the book puts it: is there a "conflict between individual and joint incentives?"

Nash Equilibrium - a solution in which one player acting alone cannot improve his or her position by changing strategy. ie without cooperation there is no way to improve one's position.

Sequential game - the situation where one repeats a game. Often the sequential outcome may differ from a single outcome, as players may learn from previous rounds.

GAME THEORY:

Social scientists have identified a number of occasions where behavior may sometimes be exemplified by a game:

politics: politicians may make certain moves, in anticipation of other politicians' actions (much of the cold war was modeled using game theory and brinksmanship ideas)

firms: pricing/innovation decisions

households: husbands and wives, parents and children

Games are relevant when there is interdependence and asymmetric information.

In order for economists to be able to analyze games, they must be aware of the pay-offs of various strategies. The book provides examples of a number of pay-off schemes starting with Figure 4.1.

In 4.1, what is Player 1's optimal choice? If Player 2 chooses L then Player 1 should choose B since he gains 6 (rather than 5). If Player 2 chooses R then Player 1 is still better off choosing B since he gets 4 rather than 3. B is dominant for 1. If the choice of a player does not depend on what the other player chooses, i.e. is always the best option, this is called a dominant strategy.

How about Player 2? The dominant strategy in this case is R.

This game is also set up as a prisoner's dilemma, since although the best strategy for both would be (T,L), if the players act independently, they will choose (B,R), which is less desirable. In other words, if each player acts alone and only looks at his/her dominant strategy, they will end up at (B,R) although (T,L) would be better for both. (Think about the following as well - is T,L a Nash equilibrium?)

How is this relevant to economics? May be particularly the case in pricing when you have firms with market power. The profit of a firm may be dependent on the pricing strategy they follow, as well as the one followed by their competition.

Why is the title of this game the prisoner's dilemma? Because the idea behind it came from the police taking two prisoners into separate rooms and telling them the following:

If you confess and your accomplice confesses, you will each get 3 years in prison.

If you confess, but your accomplice doesn't, you will get 1 year, and your accomplice will get 10.

If you do not confess, but accomplice does, you will get 10 years and accomplice gets 1.

Finally, if neither of you confesses, you go free.

So the dilemma is, you are not sure if you can count on your accomplice. If you both keep quiet that is your best solution, but if your buddy snitches, and you don't, you are in trouble. The worst you can do if you keep quiet is 10 years, but you could also go free. On the other hand the worst you can do if you snitch is 3 years, and you might get off with only 1 year. Since the potential penalty for not snitching is so great, the most likely outcome is that you both snitch, and end up with the intermediate situation. (So plea bargaining seems directly linked to game theory and prisoner's dilemma.)

Whenever there is a case where there is a "conflict between individual incentives and joint incentives" it is a prisoner's dilemma.

Now look at Figure 4.2. What can we say about that game?

In this example, Player 1 should never choose option M. This is called a dominated strategy. Thus one strategy is eliminated, but that may leave others. So while a dominant strategy tells us definitively what a firm will choose, a dominated strategy can only be eliminated. (Sort of like the difference in "Who wants to be a Millionaire" between knowing the answer with certainty and asking to have the choices narrowed down. Eliminating one or more answers doesn't necessarily give you the right answer.)

Looking again at this problem, what more can we tell. If you are player 2 and you think Player 1 is rational, what else can you conclude? Choice C only makes sense for Player 2 if Player 1 chooses M, which she is not going to do if she is rational. Now 1 has two choices left - T or B, while 2 has L and R left. But now that C has been eliminated, B is a better choice for 1 and if 1 chooses B then 2 should choose R. So from 9 possible outcomes we can determine that (B,R) will be the one reached.

In order to reach this solution we have to believe that both actors are rational and both believe the other party is rational. Further, it is necessary to have assumptions about how each player is going to act.

Going back to prisoner's dilemma, why can't the prisoners reach the solution that they both keep quiet. Does that outcome depend on rationality?

Another problem is that players may not know what the exact pay-offs facing other players are. So in order to solve a game problem you may need to make the assumption that (these conditions are not always true):

a. all parties are rational and thus choose their own optimal strategy

b. there is a clear pay-off scheme and each party has an idea of the other parties' pay-off schemes

c. parties can conjecture correctly about what other parties are likely to do and act accordingly.

A Nash equilibrium is one where one party can't change its behavior and improve its position. We will do various examples of this throughout the course.

In Figure 4.4 the Nash equilibrium is (B,R)

But not all solutions have one Nash equilibrium. For instance, look at Figure 4.5. It has multiple Nash equilibria. (We cannot be sure how this game will be solved, if players act simultaneously. If players can act sequentially, then the first player to move will have the advantage.)

While simple games are nice as examples, in reality, firms have to make repeated decisions over time. As such economists also talk about sequential games.

Complete analysis of each game included in the book:

Figure 4.1:

Dominant strategies for 1 (B) and 2 (R), Nash equilibrium, prisoner's dilemma.

Figure 4.2: No dominant strategy for 1 or 2. 1 has a dominated strategy (M). Nash equilibrium, but not a prisoner's dilemma.

Figure 4.3: No dominant strategy for 1. Dominant strategy for 2 is R. Nash equilibrium, but not a prisoner's dilemma.

Figure 4.4 (it seems the book has a typo in this example, or perhaps they are defining dominance in terms of strong dominance, since if Player 1 chooses M, Player 2 is indifferent between L, C and R. In addition, the dominated strategy for Player 1 (M) is only weakly dominated, since it is not better than all choices in T and B):

R is weakly dominant for Player 2. Player 1 does not have a dominant strategy, but does have a weakly dominated strategy - which is M. Nash equilibrium, but not a prisoner's dilemma.

Figure 4.5: No dominant or dominated, but two Nash equilibria and no prisoner's dilemma.

Paper/Rock/Scissors is an example with no dominant strategy and no dominated strategy. It has no Nash equilibria because once you know your opponent's move, you will want to change yours in order to win. It is not a prisoner's dilemma, since there is generally a clear winner/loser and so if one person becomes better off the other must become worse off. In other words cooperating never pays in this game.