Chapter 30: Game Theory

Chapter 30: Game Theory 30.1: Introduction We have now covered the two extremes perfect competition and monopoly/monopsony. In the first of these all agents are so small (or think that they are so small) that they cannot change the price; in the second of these there is one agent so dominant that he or she can choose the price. More fundamentally all the agents in these two stories do not have to think about the behaviour of others in the sense that they can predict what others will do. In this chapter and the next we consider an intermediate situation in which there are two (or more) agents and each has to think about what the others might do when they are taking decisions. More specifically, the utility of each agent not only depends on what decision they take but it also depends on what decision the other takes. To make the problem very clear, we assume in this chapter and the next that each agent must take their decisions simultaneously and independently without knowing what decision the other agent is taking. This kind of problem is known as a game and the purpose of this chapter is to analyse how people should play such games. We try to come up with some predictions which we can use in chapter 31 when we study behaviour in a duopoly. A duopoly is a situation between competition and monopoly when we have a market in which there are just two firms selling the good. 30.2: A Simple Game We consider games between two individuals: 1 and 2. We consider simultaneous play games in which the two players must take their decisions simultaneously and independently of each other. We consider games in which the payoffs to the individuals depend not only on their own decision but also the decision of the other. We give an example in figure 30.1 In this example, and in the ones that follow, we call individual 1 the row player in the sense that he or she must choose a row in the above matrix. We call individual 2 the column player in that he or she must choose a column. In this example, individual 1 has a choice between row A and row B. Individual 2 has a choice between column A and column B. The payoffs to the two players are

given in pairs in the body of the matrix the first number of the pair being the payoff the individual 1 and the second number the payoff to individual 2. As it is crucial that you are clear about this, we will explain all the payoffs in the above game. 1) If individual 1 chooses row A and individual 2 chooses column A then the payoff to individual 1 is 15 and the payoff to individual 2 is 15; 2) If individual 1 chooses row A and individual 2 chooses column B then the payoff to individual 1 is 10 and the payoff to individual 2 is 10; 3) If individual 1 chooses row B and individual 2 chooses column A then the payoff to individual 1 is 5 and the payoff to individual 2 is 5; 4) If individual 1 chooses row B and individual 2 chooses column B then the payoff to individual 1 is 0 and the payoff to individual 2 is 0; What will the players do? Recall that they must choose simultaneously and independently at the time of the choice, neither player knows what the other is choosing. If you were individual 1, what would you do? You might argue that what is best for you depends in principle on what individual 2 is doing. You might then ask yourself what is your best decision for each possible decision of individual 2? If individual 2 chooses column A then your best decision is row A (as 15 is a better payoff than 5). If individual 2 chooses column B then your best decision is row A (as 10 is a better payoff than 0). So in this game, the best decision of player 1 is to play row A independently of the choice of player 2. We say in this case that the choice of row A by player 1 is a dominant strategy for player 1 because it is best for 1 irrespective of what 2 does. What about player 2? Does he or she have a dominant strategy? In this example, yes, because row A is also best for him or her irrespective of what player 1 does. (If player 1 chooses row A the best decision for 2 is column A (since 15 is better than 10); if player 2 chooses row B the best decision for 2 is column A (since 5 is better than 0). Player 2 also has a dominant strategy in this game that of column A. So in this game we seem to have a simple prediction: individual 1 will play row A (because it is a dominant strategy for him or her) and individual 2 will play column A (because it is a dominant strategy for him or her). The outcome of the game will be (A, A) and each player will get a payoff of 15. 30.3: Nash Equilibrium As you might have anticipated, not all games are so simple. Consider game 2 in figure 30.2. Does either player have a dominant strategy? Let us check, first with player 1. 1) if player 2 chooses column A the best decision for player 1 is row A; 2) if player 2 chooses column B the best decision for player 1 is row B. So player 1 does not have a dominant strategy. What about player 2? 1) if player 1 chooses row A the best decision for player 2 is column A; 2) if player 1 chooses row B the best decision for player 2 is column A. So player 2 does have a dominant strategy that of choosing column A. This is best for player 2 independently of the decision that player 1 makes.

Let us go back to player 1. He or she does not have a dominant strategy. But he may well be able to work out that player 2 does have a dominant strategy that of choosing column A. He can then be confident that player 2 will choose column A in response to which the best decision for player 1 is row A. So in this game it seems reasonable to argue that player 2 will choose column A and having worked this out, player 1 will choose row A. The outcome of the game will be (A, A) and both players will get a payoff of 15. Moreover, in this outcome, both players would be happy in the sense that neither would want to change their choices. Notice that this is not true at any other outcome: at (A,B) player 2 would want to change; at (B,A) player 1 would want to change; at (B,B) player 2 would want to change. Because of this property of the outcome (A,A) it is called a Nash 1 Equilibrium. It is an equilibrium precisely in that sense - neither player would want to change their choice given the choice of the other player. You might ask at this point whether all games have a Nash Equilibrium - clearly game 1 above has (also (A,A)). But what about the game in figure 30.3? 1 Named after the Nobel Prize winner, John Nash.

This is a game which has two Nash Equilibria (A,A) and (B,B) 2. It might be argued that (A,A) is more likely to be the outcome in an actual game, but consider what would happen if the payoffs of '5' became large losses. If you were player 1 and were sure that player 2 would choose column A then there would be no problem but suppose you were a bit worried about player 2 and thought that he or she might choose column B. In this case playing row B might be safer for you as it avoids the possibility of incurring large losses. 30.4: Mixed Strategies There are games with no Nash Equilibria. Consider figure 30.4. Let us check: 1) (A, A) is not a Nash Equilibria because player 2 would want to change his or her decision; 2) (A, B) is not a Nash Equilibria because player 1 would want to change his or her decision; 3) (B, A) is not a Nash Equilibria because player 1 would want to change his or her decision; 4) (B, B) is not a Nash Equilibria because player 2 would want to change his or her decision. In this kind of game we need to broaden the concept of a Nash Equilibrium to other kinds of strategies. What do you think is the best strategy for each player to use? Obviously a strategy that confuses the other player if the other players knows what you are going to do they can exploit that information. And the best way to confuse? By choosing A or B at random, and, in this case, with equal probabilities. This is what is known as a mixed strategy. In this game there is an equilibrium in these mixed strategies. 2 You should pause here and make sure you understand why. (A, A) is a Nash Equilibrium because player 1 would not want to change his or her decision to play row A given that player 2 has chosen column A and because player 2 would not want to change his or her decision to play column A given that player 1 has chosen row A. (B, B) is a Nash Equilibrium because player 1 would not want to change his or her decision to play row B given that player 2 has chosen column B and because player 2 would not want to change his or her decision to play column B given that player 1 has chosen row B.

We will not consider such mixed strategy equilibria any further in this book if only for the fact that we will not make use of them. But in more advanced economics they are heavily used. They are also intellectually quite exciting. It is a pity that we cannot study them further. 30.5: The Prisoner s Dilemma This is a very famous and very important game which we shall see again in chapter 31. It is typical of many interesting economic problems, including duopoly and public goods problems. An example is presented in figure 30.5. It may prove helpful to point out the structure of this game. First this game is symmetrical though it does not need to be but it means that we can just look at the payoffs to one of the players. Let us take player 1. What is important is that in column A the payoff to player 1 is higher in row A and that in column B the payoff to player 1 is also higher in row A. But what is crucial is that the payoff to 1 in the outcome (B, B) is higher than the payoff to 1 in the outcome (A, A). Because of the symmetry of the game, this means that the outcome (B, B) Pareto dominates the outcome (A, A). In other words (B, B) is simply better for them both than (A, A). Remember this. But what is our prediction? If we look to see if either player has a dominant strategy, we see that both have: for 1 it is best to play row A irrespective of what player 2 chooses; for 2 it is best to play column A irrespective of what 1 does. The prediction that 1 will play A and that 2 will play A (because these are dominant strategies) seems to lead inexorably to the conclusion that the outcome of this game will the outcome (A, A). This is the unique Nash Equilibrium of the game. The only problem is that (A, A) is Pareto-dominated by (B, B). However, despite this attractive feature, (B, B) is not a Nash Equilibrium on the contrary both players would want to change their decision if (B, B) was the outcome: note that in this instance A is better for 1 and A is better for 2. This is the interestingly paradoxical feature of the Prisoner s Dilemma we will look at the implications for economics in chapters 31 and 33.

30.6: Nash Equilibrium when Choice is Continuous In most interesting economic applications the decision-maker usually has to choose the value of some variable often from a continuum. For example, as we will see in chapter 31, in a duopoly problem each duopolist had to choose either the quantity to produce or the price to charge and both of these can be any positive number. The ideas that have been discussed above are equally applicable to this case. Let us be more specific. Let us denote the variable of choice by q and let us continue to denote the two players by 1 and 2. So q 1 denotes the decision of player 1 and q 2 denotes the decision of player 2. The situation in which both players have a dominant strategy can be easily described. This is a situation in which the optimal decision of each player is the same irrespective of the decision of the other. We shall represent this graphically, with q 1 on the horizontal axis and q 2 on the vertical. For each player we draw a graph which represents the optimal choice of that player as a function of the decision of the other. In the case when both players have a dominant strategy we have the kind of graphs pictured in figure 30.6. In this figure the vertical line is the optimal decision for player 1 as a function of the decision of player 2. You will see that it is vertical at the value 50 which implies that the choice of 50 is optimal for player 1 irrespective of the decision of player 2. Player 1 has a dominant strategy to play 50 irrespective of what player 2 plays. The horizontal line is the optimal decision for player 2 as a function of the decision of player 1. The fact that it is horizontal at the value 50 means that it is optimal for player 2, irrespective of what player 1 chooses. Player 2 also has a dominant strategy. The outcome of this game should be clear we will end up at the intersection of the two lines with both players choosing 50. It might be useful to introduce some terminology to describe the relationship between the optimal decision of one player and the decision of the other. In economics this relationship is called the reaction function of the player. The vertical line in figure 30.6 is the reaction function of player 1 and the horizontal line the reaction function of player 2. It should be noted that this term is slightly misleading as it seems to suggest some kind of reaction, which is an irrelevant concept in a simultaneous play game, so you should remember what it means simply the optimal decision of one player as a function of the decision of the other. The reason why this is useful is that it helps us to find whether there is a Nash Equilibrium in a game. Recall that a Nash Equilibrium is an outcome for which neither player would want to change

their decisions given the decision of the other. It should be clear that any Nash equilibrium must therefore be at the intersection of the two reaction functions. Consider figure 30.7. In this is illustrated a case which we will study in chapter 31 - when the reaction functions are linear. In this example, the two functions intersect just once, at the point (33⅓, 33⅓). We can conclude that this point is the only possible Nash Equilibrium in this game: at any point off the reaction function of player 1, player 1 would want to change his or her decision; at any point off the reaction function of player 2, player 2 would want to change his or her decision. Only when we are simultaneously on both reaction functions do we have an outcome where neither player would want to change their decision. Of course we might have games in which there are 2 Nash Equilibria. An example is shown in figure 30.8. We might also have 3 (or more) Nash Equilibria. An example is presented in figure 30.9. We provide some illustrations in chapter 31.

30.7: Summary We have given a very brief overview of some aspects of game theory which will prove useful later. We have restricted ourselves to 2-player simultaneous play games - in which each player must decide what to do at the same time as the other and without knowing what the other player is choosing. We saw that some games might be simple, in that both players have a dominant strategy. A player has a dominant strategy (choice) if that choice is best irrespective of the choice of the other player. Other games might be more complicated and we introduced the important concept of a Nash Equilibrium as a possible predictor of the outcome of the game. An outcome of a game is called a Nash Equilibrium if neither player would want to change their choice given the choice of the other player. Games may have no Nash Equilibrium (in pure strategies), a unique Nash Equilibrium or multiple Nash Equilibria.. We showed that such concepts were relevant for both games with a small number of possible decisions and also those where the decision was from a continuum. For the latter we introduced the concept of a reaction function. A player s reaction function describes the optimal decision for a player as a function of the decision of the other. Any Nash Equilibrium (in pure strategies) must be at the intersection of the players reaction functions. 30.8: Playing Games Game Theory is perfect for testing in simple experiments. These are experiments that you can do with fellow students, though you are advised to have one person who keeps control of the situation (and perhaps supplies money to provide correct incentives). The games we have considered in this chapter have been simple two-person games and you are recommended to start with such games. You can always play these games with two teams, and this is, in fact, what we do when we incorporate such games into a tutorial programme. Though the word game suggests that they are not serious, that is not true the purpose of these experiments is to test Game Theory, which has many important things to say about economic behaviour. You can take virtually any game and experiment with it. In this chapter there are a number of examples and you can easily construct your own. The theory that we have been discussing in this chapter may make predictions about what will/should happen in such games, and it is therefore instructive to see if these predictions are correct. However, in some games, the theory that we have been discussing is agnostic it predicts various possibilities but does not say which of the various possibilities will emerge in practice. This happens, for example, when there are multiple Nash

equilibria. In such games, experiments are instructive in discovering which are the empirically realistic equilibria. One game that is particularly instructive with which to start is the Prisoner s Dilemma. This is interesting as there is a unique Nash equilibrium, which is supported by dominant strategies. However this Nash equilibrium is dominated by some other outcome which unfortunately is not a Nash equilibrium. It seems odd that Game Theory should predict an outcome that is dominated by another outcome. You can see what happens in practice by running a simple experiment. Consider the following. There are two players or teams. They are to play a simple simultaneous-move game in which Team 1 has two choices (1 and 2) and in which Team 2 has two choices (1 and 2). The payoffs to the teams are given in the following payoff matrix. The amounts are in pounds 3. The first number is the payoff to Team 1; the second number is the payoff to Team 2. So, for example, if Team 1 plays 2 and Team 2 plays 1, then Team 1 will get a payoff of - 2 (yes, will lose 2) and Team 2 will get a payoff of 20. Team 2's choice Team 1's choice Payoffs 1 2 1 1, 1 20,- 2 2-2, 20 6, 6 This has the classic structure of a Prisoner s Dilemma: for both Teams, the choice of 1 dominates the choice of 2: whatever the other team does, choosing 1 is best for either team. This is because 1 is better than - 2, and 20 is better than 6. As long as the game has this structure, it is a Prisoner s Dilemma and has the paradoxical prediction that both teams choose 1 and hence end up with 1 whereas they could have each ended up with 6 by both choosing 2. The objective of the teams should be to make as much (real or hypothetical) money out of playing the game 4. You might think that the outcome of the game would depend on how it was implemented, and, in particular, whether teams played it once or more than once. In fact, there are a number of alternative scenarios you could try. (1) The game is played once; (2) The game is played n times with lots of teams playing pairwise games in such a way that no two teams ever play each other more than once; (3) The game is played n times between the same two teams. Treatment (2) is called (in the experimental literature) a Strangers treatment and treatment (3) a Partners treatment. You may think that behaviour might be different in these three treatments, first 3 Ideally they should be in real pounds, and in many proper experiments they are real pounds. If you can not find anyone to sponsor your experiment, you should use hypothetical pounds, but try to act as you would if they were real. 4 In a real experiment with real money, the participants would be paid in cash their payoffs.

between (2) and (3) where strategic issues might force a difference, and second between (1) and ((2) and (3)) where learning may have an effect. An additional factor may be whether or not communication is allowed. Perhaps if the teams could discuss the possibilities they may be able to get to the dominating outcome? Perhaps more important is whether binding commitments could be made and enforced. If so, the two teams might agree after discussion both to choose 2. But how could it be enforced? There would have to be some legally binding agreement, of a type recognised by courts of law, to ensure that any team who broke the agreement would end up worse off as a consequence. In some contexts it is difficult to imagine how this might be done. And, of course, there are legal costs to be incurred in drawing up a legally binding agreement. In my tutorials I get the two teams to play the game 4 times without communication and 4 times with. I refuse to police any agreements between the teams. What happens depends upon the teams: sometimes we get cooperation, occasionally of an oscillating nature (note that alternatively playing 1 and 2 and then 2 and 1 both teams can get an average payoff of 9 per play of the game), but often the cooperation breaks down, or does not even start. You should see what happens and also see how sensitive is the outcome to the particular numbers in the payoff matrix.