Introduction to Game Theory

Transcription

1 Chapter 11 Introduction to Game Theory 11.1 Overview All of our results in general equilibrium were based on two critical assumptions that consumers and rms take market conditions for granted when they decide what to buy and what to produce. First, we assumed that all actors take market conditions as xed. Second, we assumed that the consumption and production decisions of each actor only a ect the utilities of other actors if they change what those other actors consume. We relaxed these results by allowing for rms to exert market power in our analysis of monopoly thus, the rm considers the e ect of its actions on the market conditions (e.g. the price). We also relaxed these results by allowing for externalities. In the context of economics, game theory is the study of situations that do not meet these two conditions and a game is any interaction where each person s (or organization s) actions a ect the outcomes of others. Our rst example was in our study of oligopoly. In both Cournot and Bertrand competition, there are externalities becase the action by one rm a ects the pricequantity relationship (and ultimately the pro ts) for the other rm. Further, each rm recognizes that its own action in uences the market price. These elements require a more new notion of equilibrium that incorporates a description of how rms would react to each other s actions. In this chapter, we develop a broader de nition of games that go beyond the market framework of oligopoly. Then in succeeding chapters, we consider speci c applications of game theory under the heading of "Information Economics". 273

2 11.2 Examples of Games We begin with three well-known examples to provide a sense of the considerations that pervade game theory. The rst and best-known of these games is the Prisoners Dilemma. In its original form, the Prisoners Dilemma is based on the negotiations between the police and two partners who have been caught committing a crime together. The police have enough evidence to convict both of them for a minor crime, but would like to get at least one criminal to inform on the other so that they can convict at least one of them for a major crime. The police split up the criminals and put them in di erent rooms where they cannot communicate. They then o er each the chance for a lesser sentence if he will confess and testify against the other person. The criminals, of course, have made a prior agreement with each other than neither will ever testify against the other, but now that agreement is in jeopardy. Thus, the criminals must make separate and simultaneous choices between two actions: 1) maintain the agreement and refuse the o er from the police; 2) break the agreement and accept the o er from the police. We refer to these two actions as the strategies "Cooperate" (C) and "Defect" (D), where cooperation refers to the original agreement between the criminals (and does not meet cooperating with the police). We de ne a strategy to be a complete description of how a participant will act at di erent points during the game. In the Prisoners Dilemma, each person makes a single choice of action with no knowledge of the other criminal s choice. Therefore, each person s choice of "Cooperate" or "Defect" represents that person s strategy for the game, since this single choice describes a person s actions for the entire game. We de ne a simultaneous move game to be any game where all participants choose a single action at the same time, without knowledge of the actions chosen by the others. In a simultaneous move game, an action is equivalent to a strategy. All three games discussed in this section are simultaneous move games, so we use the terms action and strategy interchangeably in discussion of these examples. 1 Table 1 shows the prison sentences the result for both criminals as a function of their strategies. The fact that each person s outcome depends on both her only strategy and on the other person s stragey underscores the interactive nature of the Prisoners Dilemma as a game. 1 An important caveat is that a strategy can involve a deliberate randomization - such as choosing one action 60% of the time and another action 40% of the time. below. Such a strategy is known as a mixed strategy and is discussed 274

3 CRIMINAL 2 Cooperate Defect CRIMINAL 1 Cooperate 2 year sentence for Criminal 1, 2 year sentence for Criminal 2 5 year sentence for Criminal 1, 1 year sentence for Criminal 2 Defect 1 year sentence for Criminal 1, 5 year sentence for Criminal 2 4 year sentence for Criminal 1, 4 year sentence for Criminal 2 Table 1: Sentences in the Prisoners Dilemma Suppose that the Bernoulli utilities for these outcomes are the same for both criminals: u(1 year sentence) = 5, u(2 year sentence) = 4, u(4 year sentence) = 1, u(5 year sentence) = 0. we can convert this table into a payo matrix for the game, as shown in Table 2. Then Cooperate Defect Cooperate 4, 4 0, 5 Defect 5, 0 1, 1 Table 2: Payo s in the Prisoners Dilemma Table 2 is known as the Normal Form for the Prisoners Dilemma that we have described above. 2 Each cell of the matrix lists a pair of utility outcomes corresponding to a particular set of actions chosen by the two criminals. In the context of game theory, we describe each participant as a "player". By convention, the rows correspond to player 1 s strategies and the columns correspond to player 2 s strategies. To distinguish between the players, we assume that player 1 is female and player 2 is male. In each pair of payo s, player 1 s utility is listed rst and player 2 s utility is listed second. For example, if player 1 choose "Defect" and player 2 chooses "Cooperate", the outcome (5, 0) indicates that player 1 receives utility of 5 and player 2 receives utility of 0. In consumer theory, we assumed that each person made choices to maximize her own utility, as represented by the solution to the Consumer Problem. In game theory, we make the same assumtion but the utility maximization problem for each player is not as well de ned as the Consumer Problem was in consumer theory. For example, player 1 s utility maximizing strategy may depend on the strategy of player 2 and further, player 1 may not be certain about what player 2 will do. 2 The Normal Form is also known as the Matrix Form or the Strategic Form representation of a game. 275

4 The Prisoners Dilemma is particularly straightforward to analyze because these complications do not a ect the result of player 1 s utility maximization problem. If player 2 chooses "Cooperate", then player 1 gets utility 5 by choosing "Defect" (1 year sentence) and utility 4 by choosing "Cooperate" (2 year sentence). Therefore, player 1 gets a lesser sentence and higher utility from "Defect" if player 2 chooses "Cooperate". In technical language, we say that player 1 s best response to "Cooperate" is "Defect". Similarly, if player 2 chooses "Defect", then player 1 gets utility 1 by choosing "Defect" (4 year sentence) and utility 0 by choosing "Cooperate" (5 year sentence). Therefore, player 1 gets a lesser sentence and higher utility from "Defect" if player 2 chooses "Defect". Once again, player 1 s best response to "Defect" is also "Defect". Thus, for either action by player 2, player 1 has the same best response, "Defect", meaning that "Defect" gives higher utility for player 1 than for player 2, regardless of what strategy player 2 chooses. 3 When one strategy for a particular player is a strict best response to all combinations of strategies for other players we say that it is a strictly dominant strategy for that player. (See Section for detailed analysis of dominant strategies.) In the Prisoners Dilemma, "Defect" is a strictly dominant strategy for each player, essentially eliminating the complexity of interaction in the Prisoners Dilemma. Although each player s action a ects the other s utility, it does not a ect the other player s maximizing decision. To maximize personal utility, each player should choose "Defect" regardless of what the other player does. We feel con dent in predicting that the outcome of the Prisoners Dilemma will be the dominant strategy outcome, ("Defect", "Defect"), meaning that each player receives a four-year sentence and utility of 1. Note that the choice of speci c utility values for the four possible combinations of strategies is not critical to the prediction that ("Defect", "Defect") will be the result of the game. The choice to "Defect" reduces one s own sentence from two years to one if the other player chooses "Cooperate" or from ve years to four if the other player chooses "Defect". Given this comparision, "Defect" will dominate "Cooperate" so long as a longer sentence gives each player less utility than a shorter sentence. The perplexing element of the Prisoners Dilemma is that the dominant strategy outcome is not 3 Player 1 s strategy "Cooperate" also provides higher utility than "Defect" even if we allow player 2 to play a "mixed strategy", choosing "Cooperate" with some probability p and "Defect" with probability 1 p. Since all utilities are Bernoulli values, each player will act to maximize expected utility when at least one player is playing a mixed strategy. In this case, player 1 s expected utility from "Cooperate" is equal to 4p + 0(1 p) or 4p, while player 1 s expected utility from "Defect" is 5p + 1(1 p) = 4p + 1. Since 4p + 1 > 4p for any p, player 1 s best response to any mixed strategy by player 2 is "Defect". 276

5 a Pareto optimum. If both players cooperate, they would receive shorter sentences than if they both defect. This outcome emphasizes the importance of externalities in games. The choice to defect reduces one s own sentence by one year, but increases the sentence to the other player by three years. Thus, each choice to defect produces an aggregate increase in two years in jail time for the two players. If both players follow their dominant strategies and defect, the net result is that they each get a sentence that is two years longer than if both cooperate. By chasing the small gain of a one-year reduction in sentence, the players ensure themselves the maximum combined sentence in total years served. Prisoners Dilemma in Oligopoly The Prisoners Dilemma can arise in a variety of more traditional economic situations. Consider the numerical example of Cournot competition from the last chapter. With demand given by p = 20 Q and constant marginal costs of 8 for each rm, the Cournot equilibrium is for each rm to produce q i = 4, for a total quantity of 8. A monopolist would limit quantity to 6, so if the Cournot rms agreed to collude, they would each produce half the monopoly quantity or 3. Consider a simultaneous move game where the players are the two rms and they make simultaneous choices to produce individual quantities of 3 ("Low") or 4 ("High"), as shown in Table 3. We assume that each player s utility is simply equal to her pro t. Low, q 2 = 3 High, q 2 = 4 Low, q 1 = 3 18, 18 15, 20 High, q 1 = 4 20, 15 16, 16 Table 3: Prisoners Dilemma Version of Cournot Competition If player 2 chooses "Low", then player 1 gets utility 20 from "High" and utility 18 from "Low". If player 2 chooses "High", then player 1 gets utility 16 from "High" and utility 15 from "Low". In either case, player 1 gets greater pro t from "High" than from "Low", so "High" is a dominant strategy for player 1. Similarly, "High" is a dominant strategy for player 2, so we predict that each player will choose "High" and that each player will receive pro t of In this game, however, total pro ts decline in the total quantity produced (assuming that total quantity is greater than the monopoly quantity of 6). Here each rm increases its own pro ts by 4 With a greater choice of strategies, Firm 1 s best response to "Low" by Firm 2 is q 1 = 3.5, not q 1 = 4. For illustrative purposes, we restrict each rm s quantity to integer levels of production for this example. 277

6 deviating from the choice of "Low" to "High", but reduces the pro ts to the other rm by a larger amount. So when both deviate from "Low" to "High", each ends up with a lower payo. 5 This game is just another version of the Prisoners Dilemma, since the dominant strategy outcome from ("High" "High") is Pareto dominated by the outcome from ("Low", "Low"). Note that this particular Prisoners Dilemma produces a bad outcome from the perspective of the rms, but a good outcome in terms of overall welfare. With a constant marginal cost of 8 and demand function p = 20 Q, a total quantity Q = 12 is socially e cient and would result in general equilibrium with many identical rms. However, with only a small number of rms, those rms exercise market power and limit production below the socially e cient level. Comparing the monopoly and duopoly outcomes, an increase in production from the monopoly level Q m = 6 to the Cournot level Q c = 8 increases societal welfare (the sum of Consumer Surplus and Producer Surplus) though it reduces the pro ts to the rms (i.e. Producer Surplus) The Stag Hunt Game A second well-known game, "Stag Hunt", is based loosely on a section from Rousseau s "A Discourse on Inequality". In this game, two hunters make simultaneous choices about what to hunt: "Stag" or "Hare". Stag is too di cult for a single hunter to catch, but hare is less valuable. Suppose that if both hunters choose to hunt stag, they work together and catch one stag for total pro ts of $400 each. If only one of them chooses to hunt stag, she goes home empty-handed. Either can choose to hunt hare on her own for a total pro t of $100. Table 4 shows the Normal form for this game, once again assuming that the utility for a hunter is equal to her total pro t. Stag Hare Stag 400, 400 0, 100 Hare 100, 0 100, 100 Table 4: Stag Hunt Neither player has a dominant strategy in this game. If player 2 chooses "Stag", then player 1 maximizes her utility by choosing "Stag", but if player 2 chooses "Hare", then player 1 maximizes 5 Unlike the original Prisoners Dilemma, the net e ect of switching from "Low" to "High" depends on the other player s strategy in this example. If player 2 chooses "Low", then a switch by player 1 from "Low" to "High" reduces aggregate pro ts by 1 unit. If player 2 chooses "High", then a switch by player 1 from "Low" to "High" reduces aggregate pro ts by 3 units. 278

7 utility by choosing "Hare". This is known as a coordination game because the players receive higher payo s if they manage to coordinate their actions with both choosing "Hare" or with both choosing "Stag" rather than having one choose "Hare" and the other choose "Stag". Thus analysis of this game requires more sophistication than did analysis of the Prisoners Dilemma. If the players cannot communicate and have no way of coordinating their actions, 6 then each might assign probabilities to the other player s choice. This is the same approach that we took in our study of uncertainty, with the di erence that the players are assigning probabilities to human choices rather than to actions that no one controls (often called "actions by nature"). If player 1 assigns probability p 2 that player 2 will choose "Stag", then player 1 s expected utility from "Stag" is 400 p 2, while player 1 s expected utility from "Hare" is 100. So player 1 should choose "Stag" if 400 p or p 2 1=4. (Note that if p 2 = 1=4 then player 1 gets expected utility 100 from "Stag" and expected utility 100 from "Hare". In this case, both of these strategies are best responses for player 1.) Similarly, player 2 should choose "Stag" if 400 p or p 1 1=4, where p 1 is the probability that player 1 will choose "Stag". Two combinations of probabilities are consistent with the solutions to these two decision problems: 1) both choose "Stag", in which case p 1 = p 2 = 1, or 2) both choose "Hare", in which case p 1 = p 2 = 0. 7 Each of these beliefs is self-con rming. If each hunter is nearly certain that the other will not show up to help hunt stag, then both will end up hunting hare. But if the hunters have con dence in each other, then both will show up to work together and they will receive the larger payo of 400 from catching stag rather than hare. This game highlights an important richness and also a de ciency of game theory. Although we have yet to give a formal de nition of equilibrium for games, it seems natural that ("Stag", "Stag") and ("Hare", "Hare") will qualify as equilibria for any reasonable de nition that we could create. So abstract analysis cannot pinpoint a particular result of this game even if we take a leap of faith to believe that the world is in equilibrium. Instead, it is necessary to consider speci c history and institutional detail to understand which of two (or more) plausible outcomes is most likely in a particular application. 6 One way that they could coordinate their actions would be to hunt stag on warm days and hunt hare on cold days. In that case, they would be able to coordinate their actions on each day even if they were not able to communicate on that particular day. The use of an external factor to coordinate play is called a "Correlated Equilibrium" and was rst described by Robert Aumann in In fact, there is a third plausible outcome to this game where the players each choose a randomized strategy. This outcome is shown in Figure 8.c and discussed in the context of mixed strategy equilibrium later in this chapter. 279

8 Rock, Paper, Scissors A third well-known game is the children s game known as "Rock, Paper, Scissors". In this game, two players each choose simultaneously between these three di erent actions. If the players choose di erent actions, then one wins and the other loses according to the following rules: "Rock" beats "Scissors", "Scissors" beats "Paper", and "Paper" beats "Rock". If both players choose the same action, then the game is a tie. Suppose that if the outcome is decisive, the loser pays a dollar to the winner and that each player s utility is simply his net payo for the game. Rock Paper Scissors Rock 0, 0-1, 1 1, -1 Paper 1, -1 0, 0-1, 1 Scissors -1, 1 1, -1 0, 0 Table 5: Rock, Paper, Scissors This game is known as a zero-sum game because if one player gains, then the other player loses an equal amount. Much of the early analysis in game theory focused on applications to battle strategies in war with the view that war was a zero-sum game. 8 If neither side has a dominant strategy in a zero-sum game, then there is no obvious outcome to the game. If either player is too predictable, then the other player can take advantage. For example, in one episode of the American television series, "The Simpsons", the Simpson children Bart and Lisa decide to write a television script together. To settle the thorny question of rst authorship, they agree to play "Rock-Paper-Scissors". Bart s immediate thought is "Good ol rock. Nothing beats that." Lisa is younger, but smarter, and her rst thought is "Poor predictable Bart. He always takes Rock". Naturally, Lisa wins the game. Rock-Paper-Scissors illustrates another challenge for game theory. Even if we assume some possibility of coordination between the players, there is no obvious outcome to the game. Thus, any de nition of equilibrium must also allow for the possibility of probabilities or randomized play Formal De nition of Games We now build a formal framework for game theory that incorporates all of these examples. 8 Thomas Schelling was one of the rst to argue that wars are not always zero-sum games. For example, both sides could gain from a well-constructed peace treaty, or both sides could lose in battles that kill civilians. 280

9 The basic components of a game are: A set of I players A description of the rules of the game. This consists of the sequence of possible decisions by players, where each decision is a choice among a set of actions the player can take. The players may also choose some or all of their actions simultaneously. The set of payo s to the players corresponding to each possible combination of strategies. These payo s are in units of Bernoulli utilities so that the players maximize "Expected Utility" in terms of known probabilities of di erent outcomes Strategies in Extensive and Normal Form Representations The examples above are all simultaneous move games and can be easily represented in the Normal Form, as described above. But games may also be sequential, with one player moving rst and the other player moving second. More complex games may also involve a series of moves in some order as well as moves by nature. These games can be represented in diagrammatic form known as a game tree or as the Extensive Form representation. Figure 1 is the extensive form representation for a game where player 1 moves rst and chooses "Up" or "Down", then player 2 observes player 1 s choice and responds by choosing "Left" or "Right". The extensive form is often called a game tree because it represents each possible sequence of play as one of many branches in a tree. Each point in a game tree where one player is called upon to act is called a decision node. In Figure 1, there are three decision nodes: node A where player 1 s starts the game by choosing an action and nodes B and C, where player 2 may be called upon to move. Even though player 2 only acts once, the game tree includes two separate decision nodes for player 2 because player 2 s action follows player 1 s action and it is not known in advance what player 1 will do. A strategy for a player is a complete contingent plan of actions for the entire game for that player. For the game in Figure 11.1, player 1 s strategy must specify her action at node A, while player 2 s strategy must specify his action at node B and his action on node C. In player 2 s strategy, each action that is speci ed for a given node is contingent upon that node being reached in the game. 9 9 The de nition of a strategy as a contingent plan of action includes a philosophical assumption that each player can anticipate every contingency in the game and identify how he would respond in that contingency. In practice, some people may say that they cannot form a plan of action for some contingencies prior to the start of the game, 281

10 1 1:pdf Figure 11.1: An Extensive Form Game A pure strategy for a player requires that player to chose an action with probability 1 at each decision node (i.e. "for certain"). Thus, there are two pure strategies for player 1 in Figure 1: "Up" and "Down". By contrast, a pure strategy for player 2 consists of one action chosen with probability 1 at node B and a separate action chosen with probability 1 at node C. There are two possible options at each decision node for player 2, so there are 2 x 2 = 4 possible pure strategies for player 2. We can represent player 2 s possible strategies in an ordered pair, where the rst entry in the pair represents the action that player 2 will take at Node B if player 1 moves "Up" and the second entry represents the action that player 2 will take at Node C if player 1 moves "Down". The four possible pure strategies for player 2 for the game in Figure 1 are "Left, Left", "Left, Right", "Right, Left", "Right, Right". Table 6a shows the actions taken by the players in the extensive form game in Figure 1 as a function of their pure strategies. Table 6a would be a normal form representation of this game, except that it lists actions rather than payo s in each cell. Table 6b is the normal form representation for this game. particularly shocking ones. (e.g. "I can t tell if I ll want to take revenge if you betray me, because it s unthinkable to me that you would betray me.") A separate practical di culty is that in complicated games (e.g. chess), it may be impossible for each player to contemplate all possible contingencies. 282

11 Left, Left Left, Right Right, Left Right, Right Up Up, Left Up, Left Up, Right Up, Right Down Down, Left Down, Right Down, Left Down, Right Table 6a: Actions Chosen by Players in Figure 1 as a Function of Their Strategies Left, Left Left, Right Right, Left Right, Right Up 1, 4 1, 4 4, 1 4, 1 Down 3, 2 2, 3 3, 2 2, 3 Table 6b: Normal Form Representation for the Extensive Form Game in Figure 1 In some cases, a player may be called upon to act, but may not know precisely which node in the tree has been reached. For example, if player 2 does not observe player 1 s action at node A in Figure 1, then he would not know whether he was making a choice at node B or at node C. We describe this possibility with the use of an information set, which is a set of decision nodes for a particular player who cannot distinguish among these nodes at the time she must choose an action. Note that the set of possible actions must be the same at each decision node in an information set, for otherwise, the player could use the set of possible actions to distinguish among at least some of the nodes in the information set. Figure 2 shows the extensive form representation for a simultaneous move game where player 1 chooses either "Up" or "Down" and player 2 chooses either "Left" or "Right". Figure 2 is identical to Figure 1 except for the addition of the dotted line connecting nodes B and C. This dotted line indicates that these two nodes are in the same information set, meaning that player 2 cannot tell if he is at node B or at node C when he is called upon to move. From a strategic standpoint, information is more important that the exact timing of decisions. If player 1 chooses a strategy at 11:30 and player 2 chooses a strategy at 11:45, then the game is literally sequential. But if player 2 does not observe player 1 s choice prior to 11:45, then from player 2 s perspective, the game might as well be simultaneous since player 2 has exactly the same information (none) about player 1 s choice at 11:45 as when the game started. For this reason, we use the same extensive form representation for a simultaneous move game and a sequential move game where player 1 moves before player 2 but player 2 does not observe player 1 s move. 283

12 2 2:pdf Figure 11.2: Extensive Form for a Simultaneous Move Game Mixed and Behavioral Strategies A mixed strategy is a randomization, where a player puts positive probability on at least two of her strategies. In a game where the players have just two strategies each, such as Stag Hunt, any mixed strategy can be described by a single probability: if player 1 plays "Stag" with probability p, then she must play "Hare" with the remaining probability 1 p. In more complicated games, a mixed strategy is a vector of probabilities, where the probabilities sum to 1 and each element of the vector represents the probability of playing a particular pure strategy in the game. For example, in Table 6b, a mixed strategy for player 2 is a vector of four probabilities (p 1; p 2; p 3; p 4 ) where p 1 + p 2 + p 3 + p 4 = 1 and p 1 represents the probability of pure strategy ("Left", "Left"), p 2 represents the probability of pure strategy ("Left", "Right"), p 3 represents the probability of pure strategy ("Right", "Left"), and nally p 4 represents the probability of pure strategy ("Right", "Right") Since the sum of probabilities for player 2 s pure strategies must add to 1, the values of any three of the strategies are su cient to determine the fourth probability. For this reason, we say that there are three degrees of freedom in determining player 2 s mixed strategy among four possible pure strategies. 284

13 285

14 Another way of representing randomization for player 2 is to allow player 2 to randomize between strategies at each node. A node-by-node list of randomizations for a player is known as a behavioral strategy. Returning to the game in Figure 11.1, a behavioral strategy for player 2 consists of two probabilities p B and p C, where p B is the probability that player 2 chooses "Left" at node B and p C is the probability that player 2 chooses "Left" at node C, as shown in Figure We do not emphasize the distinction between behavioral and mixed strategies, because it is possible to convert a behavioral strategy into a mixed strategy with equivalent probablities for player 2 s action at each node. To convert a behavioral strategy in Figure 11.3 to a mixed strategy, assume that player 2 actually chooses an action at both node B and node C even though only one of those nodes can be reached in the game. Further assume these choices are statistically independent so that P(Left, Left) = p B p C. 11 Then the behavioral strategy (p B ; p C ) corresponds to the mixed strategy (p 1; p 2; p 3; p 4 ) where p 1 = p B p C, p 1 = p B (1 p C ); p 3 = (1-p B ) p C ; p 4 = (1-p B ) (1 p C ). Note that P (Left j Up) = p 1 + p 2 = p B p C + p B (1 p C ) = p B and similarly that P (Left j Down) = p C. Thus, this mixed strategy produces the same probabilities for player 2 s actions as the behavioral strategy, so the two are truly equivalent. Since it is generally easier to analyze the best response to a mixed strategy than the best response to a behavioral strategy, and there is an equivalence between mixed and behavioral strategies, we will concentrate on mixed strategies in further discussion Equivalence of Extensive and Normal Form Representations Our method for converting the extensive form game in Figure 1 to the normal form game in Table 6b can be generalized into an algorithm to convert any nite extensive form game into a normal form game. (Any game with a nite number of players and a nite number of actions at each information set is a nite game.). To identify the pure strategies for each player, rst identify the number of information sets at which that player could be called upon to move. Then create a set of strategies where each strategy is a vector containing an entry with an action for each information set. For example, if there are ve information sets where player 1 could be called upon to move 11 If we assume that there is statistical correlation between the resolution of uncertainty for player 2 s actions at nodes B and C, we would still be able to convert player 2 s behavioral strategy into a mixed strategy, but we would identify a di erent mixed strategy than when these actions are statistically independent. However, this mixed strategy would have the same probability of each action at each node and so player 1 s best response would be the same in both cases. 286

15 and three possible actions at each information set, then each strategy would be a vector with ve entries and there would be a total of 3 5 pure strategies for player 1. Naturally, as the game becomes more complex, the number of possible strategies for player 1 grows exponentially. It may not be practical to represent a game with many information sets for each player in the normal form, but it is at least theoretically possible to do so. In some cases, our procedure for converting an extensive form game to a normal form game may appear to include some redundant strategies. For example, Figure 4 shows a case where player 1 may be called upon to move twice, but only if his rst move is "Continue". There are four possible strategies for player 1: ("Continue", "Up"), ("Continue", "Down"), ("Stop", "Up"), ("Stop", "Down"), but both ("Stop", "Up") and ("Stop", "Down") both end the game immediately. Despite the fact that the strategies ("Stop", "Up") and ("Stop", "Down) may seem equivalent, it is important to include both of them in the analysis of the game, as indicated by the following logic. 287

16 Player 1 can end the game immediately by choosing "Stop" at node A for a payo of (2, 0). Player 1 s payo from "Continue" at node A depends on player 2 s (anticipated) action at node B. If player 2 would choose "Left" at node B, then player 1 s would get a payo of 5 from ("Continue", "Up") and "3 from ("Continue", "Down"). Either of these outcomes gives player 1 a higher payo than if player 1 chooses "Stop" at node A. But if player 2 would choose "Right" at node B, then player 1 does better by choosing "Stop" than "Continue" at node A. To this point, we have not distinguished in this analysis between ("Stop", "Up") and ("Stop", "Right"). This distinction is paramount for player 2 s choice of action at node B. If player 2 is called upon to move at node B, he gets a higher payo from "Left" if player 1 would choose "Down" at node C and a higher payo from "Right" if player 1 would choose "Up" at node C. That is, the di erence between ("Stop", "Up") and ("Stop", "Down") determines whether player 2 should choose "Continue" or "Stop" at node B. If we merge player 1 s strategies ("Stop", "Up") and ("Stop", "Right") into the single strategy "Stop", then it would be impossible to identify player 2 s utility maximizing strategy at node B in response to player 1 s strategy, "Stop". Then in turn, it would be impossible for player 1 to determine if "Stop" is her utility maximizing strategy at node A if player 2 s action at node B is not speci ed. For this reason, we include both ("Stop", "Up") and ("Stop", "Down") as distinct pure strategies for player 1 in the analysis of the extensive form for this game. However, a di erent convention applies to the representation of this game in the normal form. Both strategies ("Stop", "Up") and ("Stop", "Down") yield the same outcome (2, 0), regardless of player 2 s strategy, as shown in Table 6c. Left Right Continue, Up 5, 1 1, 2 Continue, Down 3, 3 1, 2 Stop, Up 2, 0 2, 0 Stop, Down 2, 0 2, 0 Table 6c: Full Normal Form for the Extensive Form Game in Figure 4 288

17 In the normal form, there is no reason or way to distinguish between ("Stop, Up") and ("Stop, Down"). Therefore, it is appropriate to combine these two strategies in the reduced normal form of the game, as shown in Table 6d. But this distinction in the conventional representation of the game four pure strategies for player 1 in the extensive form of the game, but only three pure strategies for player 1 in the reduced normal form of the game - suggests that a subtle di erence between the extensive and normal form representations of games where at least one player acts more than once. This distinction in uences the analysis of dynamic games - as discussed later in these notes under the headings of Subgame Perfect equilibrium and Perfect Bayesian equilibrium. Left Right Continue, Up 5, 1 1, 2 Continue, Down 3, 3 1, 2 Stop 2, 0 2, 0 Table 6d: Reduced Normal Form for the Extensive Form Game in Figure 4 Converting a Normal Form Game into an Extensive Form Game There are multiple ways to convert a normal form representation into an extensive form game with the same set of strategies and payo s. For example, the extensive form games in both Figure 11.1 and Figure11 5 correspond to the same normal form game in Table 6b. The simplest method for converting a normal form to an extensive form game is simply to condense all of the actions for each player into a single choice of moves - including all possible pure strategies as separate actions at a single information set for that player. This is an awkward choice that meets the literal de nition of the extensive form, but takes the spirit of a normal form (simultaneous move) game. In addition, brute force conversion from the normal form to the extensive form may suppress important strategic considerations. For example, the representation in Figure 1 highlights player 2 s ability to observe player 1 s move and to respond optimally to it, but the representation in Figure 5 obscures player 2 s strategic advantage in the game. The equivalence of the extensive form games in Figures 1 and 5 is based on the assumption that the players make complete contingent plans prior to the start of the game. Even with this assumption, it is natural to prefer the representation in Figure 1, which accurately depicts the series of moves in the game, to the representation in Figure 5 which suppresses this information. For our purpose of exposition here, it is only important that it is possible to represent a normal form game in the extensive form and vice versa. Now that we have established this possibility, we 289

18 4 5:pdf Figure 11.3: Second Extensive Form for Simultaneous Move Game have the freedom to represent any game in either form in further discussion Moves by Nature A game may include elements of uncertainty that are common to all players, and some or all of these uncertainties may be resolved during the game. For example, in a game that involves negotiation between a venture capitalist and an entrepreneur, there might be a preliminary report about the entrepreneur s pro tability during the course of the negotiation. In game theory, we describe the resolution of uncertainty as a move by nature, where nature is modeled as a player that acts probabilistically rather than to achieve a particular objective. In the simplest case of uncertainty caused by a move by nature, two players do not know which of two simultaneous move games that they are playing. For example, players 1 and 2 may not know whether they are playing the game in Table 7a or the game in Table 7b. 290

19 Left, Right Up 4, 0 1, 2 Down 3, 4 2, 1 Table 7a: One of Two Possible Games Left, Right Up 2, 1 3, 4 Down 1, 2 4, 0 Table 7b: The Second of Two Possible Games Suppose that the players choose their strategies simultaneously, and then only learn which of the two games they were actually playing when they learn their payo s. This uncertainty can be represented in the extensive form as a move by nature that takes place after the moves by each player. It can also be represented in the extensive form as a move by nature that takes place prior to the moves by the two players. So long as we assume that the players know the probability,, that they are playing game 1 (so that they are playing game 2 with probability 1 ), then we can incorporate the move by nature into their normal form payo s. If the players play ("Up", "Left") for example, they receive payo s of (4,0) from Game 1 with probability and they receive a payo of (2; 1) from Game 2 with probability 1. Since their payo s in each game are assumed to be Bernoulli utilities, the players act to maximize their expected utilities when uncertainties (either due to moves by nature or mixed strategies by other players) are involved. Combining the two possible payo s for ("Up", "Left"), the expected utilities for the two players are are (4 + 2(1 ); 1(1 )) or (2+2, 1-). Table 7c represents the normal form for the probability weighted combination of these two possible games. Left, Right Up 2+2, 1-3-2; Down 1+2, , Table 7c: Normal Form Incorporating a Move by Nature into Expected Utility In cases where there are moves by nature, but none of the players observe those moves, then uncertainty about nature s actions can simply be incorporated into the (expected) payo s of the game, as shown by example in Table 7c above. When some, but not all players observe one of 291

20 nature s moves, then the game involves asymmetric information. We discuss these games separately in the section on Bayes-Nash equilibrium below and subsequently in the chapters on Information Economics Solution Concepts in Game Theory Now that we have a method for representing games in the normal and extensive forms, we would like to be able to make robust predictions about how economically rational actors might play those games. We call each rule for predicting the outcome of a game a solution concept. Almost all solution concepts are based on the idea of optimal action / best response by each player; solution concepts di er by requiring more or less restrictive assumptions about what other players will do Iterative Solution Methods The simplest concept is that of Dominance, which we explored in the context of the Prisoners Dilemma above. We say that strategy A strictly dominates strategy B if strategy A gives a strictly higher payo than does strategy B for each possible combinations of strategies by other players. We say that strategy A weakly dominates strategy B if strategy A gives at least as high a payo as strategy B for each possible combinations of strategies by other players and strategy A gives a strictly higher payo than strategy B for some possible combination(s) of strategies by other players. 12 Dominance generalizes to a procedure of elimination of strategies by iterated strict dominance. Roughly, iterated strict dominance says that if A is preferred to B in all but a ridiculous set of circumstances, then we should select A over B. The de nition of the ridiculous set of circumstances are those in which other players select clearly faulty strategies. In Table 8a below, Middle dominates Right for player 2, but there are no other dominated strategies. Left Middle Right Up 4,-4 1, 4 0, -3 Down 5, 3 2, 2-1, -2 Table 8a: A Normal Form Game with One Dominant Strategy 12 If strategies A and B give the same payo for each possible combination of strategies for other players, then these strategies are equivalent. We could say that A and B weakly dominate each other in this case. 292

21 As this example illustrates, Dominance emphasizes the strategy that is dominated. In this case, we can eliminate "Right" since player 2 would always get higher utility with "Middle" than with "Right". However, the fact that "Middle" rather than "Left" is the strategy that dominates "Right" does not necessarily mean that there is any reason to prefer "Middle" to "Left" in the choice of the two remaining strategies for player 2. Instead, we proceed by eliminating "Right" and then examining the resulting 2x2 game shown in Table 8b for dominant strategies. Left Middle Up 4, -4 1, 4 Down 5, 3 2, 2 Table 8b: The Normal Form Game with Removal of One Dominated Strategy After the elimination of Right, Down dominates Up. So we can eliminate Up. This leaves only one strategy for player 1, Down, and two strategies for player 2, as shown in Table 8c. Left Middle Down 5, 3 2, 2 Table 8c: The Normal Form Game with Removal of Another Dominated Strategy Comparing the payo s for player 2, Left gives the higher payo and is the best response to Down meaning that Left dominates Middle after the elimination of Up for player 1. Thus, we can eliminate Middle, leaving only one strategy for each player, Down for player 1 and Left for player 2, as shown in Table 8d. Left Down 5, 3 Table 8d The Normal Form Game with Removal of Another Dominated Strategy With only one strategy left for each player, we predict the outcome of the game to be (Down, Left). If iterated dominance yields an exact prediction for each player s strategy, then we say that the game is dominance solvable. (Note that any game with a dominant strategy for each player, 293

22 such as the Prisoners Dilemma, is de ned to be solvable by iterated dominance even though the dominance procedure does not have to be iterated to produce this solution.) Intuitively, when iterated dominance yields a speci c prediction about the outcome of a game, each player is playing a best response to all possible combinations of strategies by other players, excluding implausible strategies an implausible strategy means any strategy excluded by the iterated dominance procedure. For example, examining Table 8a, player 1 should only consider "Up" if player 2 is expected to play "Right", but "Right" is a dominated strategy. This logic, which corresponds to the rst two steps in iterated elimination of dominated strategies, provides a strong case for the prediction of ("Down", "Left"). Dominance is the most stringent criterion in common use in game theory for predicting the outcome of a game. The downfall to this criterion is that many games, such as "Stag Hunt" and "Rock-Paper-Scissors", are not dominance solvable. In general, if iterated (strict) dominance identi es a solution to a game, we tend to believe that sophisticated players will follow the dominant strategy outcome particularly if only one or two stages of elimination are required to identify the solution. Even though the dominant strategy outcome in the Prisoners Dilemma involves regrettably little cooperation, we still believe that both sides will defect in any Prisoners Dilemma. However, there are some extreme cases where the a dominant strategy outcome is unlikely to be played. Table 8c illustrates one such case. Left Middle Right Up 4,-4 1, 4 0, -3 Down 5, 3 2, 2-1,000,000, -2 Table 8e: A Game where Players May Deviate from Dominance Predictions Table 8e repeats the game from Table 8a with a single change in payo s: player 1 s payo from ("Down", "Right") has been changed from -1 to -1,000,000. This change in payo s does not a ect the results of iterated dominance. As before, "Middle" dominates "Right", then "Down" dominates "Up" after the elimination of "Right", and nally "Left" dominates "Middle" after the elimination of "Right" and "Up". The iterated dominance solution to this game remains ("Down", "Left"). But now the possibility of payo -1,000,000 for player 1 weakens the prediction that the players will play ("Down", "Left"). The argument in favor of "Down" is that with these payo s, player 2 would not play "Right". Assuming that player 2 would not play "Right", player 1 increases her payo by 1 with the choice 294

23 of "Down" instead of "Up". However, if there is even a small probability that player 2 would play "Right", the possibility of a payo of -1,000,000 from ("Down", "Right"), could in uence player 1 to choose "Up". For example, if P(Right) = 1 / 1,000,000, then player 1 would get very slightly higher expected utility from "Down" instead of "Up". In practice, we suspect that very few people would choose "Down" as player 1 in this game. There are at least three reasons that player 2 could play "Right": 1) this assessment of player 2 s payo s is incorrect and in fact, player 2 would bene t rather than lose by playing "Right"; 2) player 2 could make a mistake and play "Right" even with the knowledge that this is a dominated strategy; 3) player 2 might derive positive utility from imposing a disastrous result on player 1. Considering any one of these reasons would likely be su cient for player 1 to avoid playing "Down". This example highlights several assumptions that are necessary to translate the reasoning from iterated dominance into a prescription for play. Speci cally, iterated dominance requires each player not only to make sophisticated calculations, but also to assume that other players will make those same sophisticated calculations. This requirement is known as common knowledge of economic rationality. It is not a trivial assumption, but it is standard in game theory. In addition, iterated dominance requires common knowledge of the payo s in the game - each player must be certain of the payo s for other players in the game. Each additional step to eliminate strategies requires one more level of certainty in terms of common knowledge. For player 1 to choose "Up" in place of "Down", she must be certain of player 2 s payo s and also that player 2 is sophisticated enough to complete one round of dominance reasoning to eliminate "Down". For player 2 then to eliminate "Middle", he must be certain of player 1 s payo s, that player 1 knows player 2 s payo s and will conclude that player 2 to eliminate "Right", and that player 1 will be sophisticated enough to choose "Down" after concluding that player 2 will not play "Right". Common knowledge is often represented as a chain. Here from player 2 s perspective, the choice to eliminate "Middle" is predicated on reasoning of the form, "I know that you know my payo s and that I am economically rational." The assumption of common knowledge in game theory allows for reasoning of this form of any length, thus allowing any number of steps necessary to identify a solution using the iterated dominance procedure. However, it is clear that as we add more stages of reasoning, the common knowledge assumption becomes more burdensome. At least for complicated applications of iterated dominance, it may be desirable to ask oneself if the prediction relies too heavily on common knowledge to be believable in practice. 295

24 Nash Equilibrium The greatest defect of the dominant strategy criterion for solving games is that many games cannot be solved by iterated dominance. The more general concept of Nash equilibrium applies to games that do not have dominant strategy outcomes. A Nash equilibrium is a list of strategies for the players in a game such that each player s strategy is a best response to the strategies of the other players. If a game can be solved by iterated dominance, then the solution is a Nash equilibrium. 13 So, Nash equilibrium is a strictly weaker requirement than iterated dominance. We consider two types of Nash equilibrium outcomes in turn. Pure Strategy Nash Equilibrium In a pure strategy Nash equilibrium, both players select a strategy with probability 1 (i.e. no mixed / randomized strategies). It is usually easiest to analyze a game in the normal form to nd pure strategy Nash equilibria. Nash equilibrium. We use the game in Table 9a to illustrate how to nd a pure strategy Left Middle Right Up 2, 5 3, 4 7, 8 Down 1, 6 6, 7 4, 2 Table 9a: A Normal Form Game One way to identify pure strategy Nash equilibria when neither player has a dominant strategy is to "Guess and Verify". There are six possible combinations of pure strategies, so it is possible to check combination individually to see if it produces a Nash equilibrium. For example, the combination ("Up", "Left") is not a Nash equilibrium because player 2 s best response to "Up" is "Right". That is, in the combination ("Up", "Left"), player 1 is playing a best response to player 2 s strategy, but player 2 is not playing a best response to player 1 s strategy. Exhaustive use of 13 We don t prove this formally here, but the reasoning is straightforward. If a game can be solved by iterated dominance, then the solution is clearly a Nash equilibrium among all strategies that remain after the removal of dominated strategies at early stages of reasoning. So the only way that the iterated dominance solution could fail to be a Nash equilibrium would be if one player could improve her utility by switching to a strategy that was eliminated in an earlier stage of reasoning. But if this is possible, that strategy should not have been removed at any earlier stage - meaning that in fact, the iterated dominance solution must in fact be a Nash equilibrium. 296