Game Theory: Noncooperative 2-Player Games, Guías, Proyectos, Investigaciones de Matemáticas

¡Descarga Game Theory: Noncooperative 2-Player Games y más Guías, Proyectos, Investigaciones en PDF de Matemáticas solo en Docsity! A Brief Introduction to the Basics of Game Theory Matthew O. Jackson, Stanford University I provide a (very) brief introduction to game theory. I have developed these notes to provide quick access to some of the basics of game theory; mainly as an aid for students in courses in which I assumed familiarity with game theory but did not require it as a prerequisite. Of course, the material discussed here is only the proverbial tip of the iceberg, and there are many sources that offer much more complete treatments of the subject.1 Here, I only cover a few of the most fundamental concepts, and provide just enough discussion to get the ideas across without discussing many issues associated with the concepts and approaches. Fuller coverage is available through a free on-line course that can be found via my website: http://www.stanford.edu/∼jacksonm/ The basic elements of performing a noncooperative 2 game-theoretic analysis are (1) framing the situation in terms of the actions available to players and their payoffs as a function of actions, and (2) using various equilibrium notions to make either descriptive or 1 For graduate-level treatments, see Roger Myerson’s (1991) Game Theory: Analysis of Conflict, Cam- bridge, Mass.: Harvard University Press; Ken Binmore’s (1992) Fun and Games, Lexington, Mass.: D.C. Heath; Drew Fudenberg and Jean Tirole’s (1993) Game Theory, Cambridge, Mass.: MIT Press; and Martin Osborne and Ariel Rubinstein’s (1994) A Course in Game Theory, Cambridge, Mass.: MIT Press. There are also abbreviated texts offering a quick tour of game theory, such as Kevin Leyton-Brown and Yoav Shoham’s (2008) Essentials of Game Theory, Morgan and Claypool Publishers. For broader readings and undergraduate level texts, see R. Duncan Luce and Howard Raiffa (1959) Games and Decisions: Introduction and Critical Survey; Robert Gibbons (1992) Game Theory for Applied Economists; Colin F. Camerer (2003) Behavioral Game Theory: Experiments in Strategic Interaction; Martin J. Osborne (2003) An Introduction to Game Theory; Joel Watson (2007) Strategy: An Introduction to Game Theory; Avinash K. Dixit and Barry J. Nalebuff (2010) The Art of Strategy: A Game Theorist’s Guide to Success in Business and Life; Joseph E. Harrington, Jr. (2010) Games, Strategies, and Decision Making, Worth Publishing. 2“Noncooperative game theory” refers to models in which each players are assumed to behave selfishly and their behaviors are directly modeled. “Cooperative game theory,” which I do not cover here, generally refers to more abstract and axiomatic analyses of bargains or behaviors that players might reach, without explicitly modeling the processes. The name “cooperative” derives in part from the fact that the analyses often (but not always) incorporate coalitional considerations, with important early analyses appearing in John von Neumann and Oskar Morgenstern’s 1944 foundational book “Theory of Games and Economic Behavior.” 1 Electronic copy available at: https://ssrn.com/abstract=1968579 prescriptive predictions. In framing the analysis, a number of questions become important. First, who are the players? They may be people, firms, organizations, governments, ethnic groups, and so on. Second, what actions are available to them? All actions that the players might take that could affect any player’s payoffs should be listed. Third, what is the timing of the interactions? Are actions taken simultaneously or sequentially? Are interactions repeated? The order of play is also important. Moving after another player may give player i an advantage of knowing what the other player has done; it may also put player i at a disadvantage in terms of lost time or the ability to take some action. What information do different players have when they take actions? Fourth, what are the payoffs to the various players as a result of the interaction? Ascertaining payoffs involves estimating the costs and benefits of each potential set of choices by all players. In many situations it may be easier to estimate payoffs for some players (such as yourself) than others, and it may be unclear whether other players are also thinking strategically. This consideration suggests that careful attention be paid to a sensitivity analysis. Once we have framed the situation, we can look from different players’ perspectives to analyze which actions are optimal for them. There are various criteria we can use. 1 Games in Normal Form Let us begin with a standard representation of a game, which is known as a normal form game, or a game in strategic form: • The set of players is N = {1, . . . , n}. • Player i has a set of actions, ai, available. These are generally referred to as pure strategies.3 This set might be finite or infinite. • Let a = a1 × · · · × an be the set of all profiles of pure strategies or actions, with a generic element denoted by a = (a1, . . . , an). 3The term “pure” indicates that a single action is chosen, in contrast with “mixed” strategies that I discuss below, in which there is a randomization over actions. 2 Electronic copy available at: https://ssrn.com/abstract=1968579 ui(ai, a−i) ≥ ui(a ′ i, a−i) for all a′i and all a−i ∈ a−i. A strategy is a strictly dominant strategy if the above inequality holds strictly for all a′i 6= ai and all a−i ∈ a−i. Dominant strategies are powerful from both an analytical point of view and a player’s perspective. An individual does not have to make any predictions about what other players might do, and still has a well-defined best strategy. In the prisoners’ dilemma, it is easy to check that each player has a strictly dominant strategy to defect—that is, to confess to the police and agree to testify. So, if we use dominant strategies to predict play, then the unique prediction is that each player will defect, and both players fare worse than for the alternative strategies in which neither defects. A basic lesson from the prisoners’ dilemma is that individual incentives and overall welfare need not coincide. The players both end up going to jail for 2 years, even though they would have gone to jail for only 1 year if neither had defected. The problem is that they cannot trust each other to cooperate: no matter what the other player does, a player is best off defecting. Note that this analysis presumes that all relevant payoff information is included in the payoff function. If, for instance, a player fears retribution for confessing and testifying, then that should be included in the payoffs and can change the incentives in the game. If the player cares about how many years the other player spends in jail, then that can be written into the payoff function as well. When dominant strategies exist, they make the game-theoretic analysis relatively easy. However, such strategies do not always exist, and then we can turn to notions of equilibrium. 1.2 Nash Equilibrium A pure strategy Nash equilibrium4 is a profile of strategies such that each player’s strategy is a best response (results in the highest available payoff) against the equilibrium strategies of the other players. 4The concept is named after John Nash, who provided the first existence proof in finite games: Nash, J.F. (1951) Non-Cooperative Games, Annals of Mathematics 54:286295. On occasion it is also referred to as Cournot–Nash equilibrium, with reference to Antoine Augustin Cournot, who in the 1830’s first developed such an equilibrium concept in the analysis of oligopoly (a set of firms in competition with one another) : Cournot (1838) Recherches sur les principes mathematiques de la theorie des richesses, translated as: Researches into the Mathematical Principles of the Theory of Wealth, New York: Macmillan (1897). 5 Electronic copy available at: https://ssrn.com/abstract=1968579 A strategy ai is a best reply, also known as a best response, of player i to a profile of strategies a−i ∈ a−i for the other players if ui(ai, a−i) ≥ ui(a ′ i, a−i) for all a′i. A best response of player i to a profile of strategies of the other players is said to be a strict best response if it is the unique best response. A profile of strategies a ∈ A is a pure strategy Nash equilibrium if ai is a best reply to a−i for each i. That is, a is a Nash equilibrium if ui(ai, a−i) ≥ ui(a ′ i, a−i) for all i and a′i. This definition might seem somewhat similar to that of dominant strategy, but there is a critical difference. A pure strategy Nash equilibrium only requires that the action taken by each agent be best against the actual equilibrium actions taken by the other players, and not necessarily against all possible actions of the other players. A Nash equilibrium has the nice property that it is stable: if each player expects a to be the profile of actions played, then no player has any incentive to change his or her action. In other words, no player regrets having played the action that he or she played in a Nash equilibrium. In some cases, the best response of a player to the actions of others is unique. A Nash equilibrium such that all players are playing actions that are unique best responses is called a strict Nash equilibrium. A profile of dominant strategies is a Nash equilibrium but not vice versa. To see another illustration of Nash equilibrium, consider the following game between two firms that are deciding whether to advertise. Total available profits are 28, to be split between the two firms. Advertising costs a firm 8. Firm 1 currently has a larger market share than firm 2, so it is seeing 16 in profits while firm 2 is seeing 12 in profits. If they both advertise, then they will split the market evenly and get 14 in base profits each, but then must also pay the costs of advertising, so they receive see net profits of 6 each. If one advertises while the other does not, then the advertiser captures three-quarters of the market (but also pays for advertising) and the non-advertiser gets one-quarter of the market. (There 6 Electronic copy available at: https://ssrn.com/abstract=1968579 are obvious simplifications here: just considering two levels of advertising and assuming that advertising only affects the split and not the total profitability.) The net payoffs are given in the Table 3. Table 3: An Advertising Game Firm 2 Not Adv Firm 1 Not 16, 12 7, 13 Adv 13, 7 6, 6 To find the equilibrium, we have to look for a pair of actions such that neither firm wants to change its action given what the other firm has chosen. The search is made easier in this case, since firm 1 has a strictly dominant strategy of not advertising. Firm 2 does not have a dominant strategy; which strategy is optimal for it depends on what firm 1 does. But given the prediction that firm 1 will not advertise, firm 2 is best off advertising. This forms a Nash equilibrium, since neither firm wishes to change strategies. You can easily check that no other pairs of strategies form an equilibrium. While each of the previous games provides a unique prediction, there are games in which there are multiple equilibria. Here are three examples. Example 1 A Stag Hunt Game The first is an example of a coordination game, as depicted in Table 4. This game might be thought of as selecting between two technologies, or coordi- nating on a meeting location. Players earn higher payoffs when they choose the same action than when they choose different actions. There are two (pure strategy) Nash equilibria: (S, S) and (H, H). This game is also a variation on Rousseau’s “stag hunt” game.5 The story is that two hunters are out, and they can either hunt for a stag (strategy S) or look for hares (strategy H). Succeeding in getting a stag takes the effort of both hunters, and the hunters are separated 5To be completely consistent with Rousseau’s story, (H, H) should result in payoffs of (3, 3), as the payoff to hunting for hare is independent of the actions of the other player in Rousseau’s story. 7 Electronic copy available at: https://ssrn.com/abstract=1968579 following simple variation on a penalty kick in a soccer match. There are two players: the player kicking the ball and the goalie. Suppose, to simplify the exposition, that we restrict the actions to just two for each player (there are still no pure strategy equilibria in the larger game, but this simplified version makes the exposition easier). The kicking player can kick to the left side or to the right side of the goal. The goalie can move to the left side or to the right side of the goal and has to choose before seeing the kick, as otherwise there is too little time to react. To keep things simple, assume that if the player kicks to one side, then she scores for sure if the goalie goes to the other side, while the goalie is certain to save it if the goalie goes to the same side. The basic payoff structure is depicted in Table 7. Table 7: A Penalty-Kick Game. Goalie L R Kicker L -1, 1 1, -1 R 1, -1 -1, 1 This is also the game known as “matching pennies.” The goalie would like to choose a strategy that matches that of the kicker, and the kicker wants to choose a strategy that mismatches the goalie’s strategy.7 It is easy to check that no pair of pure strategies forms an equilibrium. What is the solution here? It is just what you see in practice: the kicker randomly picks left versus right, in this particular case with equal probability, and the goalie does the same. To formalize this observation we need to define randomized strategies, or what are called mixed strategies. For ease of exposition suppose that ai is finite; the definition extends to infinite strategy spaces with proper definitions of probability measures over pure actions. 7For an interesting empirical test of whether goalies and kickers on professional soccer teams randomize properly, see Chiappori, Levitt, and Groseclose (2002) Testing Mixed-Strategy Equilibria When Players Are Heterogeneous: The Case of Penalty Kicks in Soccer, American Economic Review 92(4):1138 - 1151; and see Walker and Wooders (2001) Minimax Play at Wimbledon, American Economic Review 91(5):1521 - 1538. for an analysis of randomization in the location of tennis serves in professional tennis matches. 10 Electronic copy available at: https://ssrn.com/abstract=1968579 A mixed strategy for a player i is a distribution si on ai, where si(ai) is the probability that ai is chosen. A profile of mixed strategies (s1, . . . , sn) forms a mixed-strategy Nash equilibrium if ∑ a ∏ j sj(aj) ui(a) ≥ ∑ a−i ∏ j 6=i sj(aj) ui(a ′ i, a−i) for all i and a′i. So a profile of mixed strategies is an equilibrium if no player has some strategy that would offer a better payoff than his or her mixed strategy in reply to the mixed strategies of the other players. Note that this reasoning implies that a player must be indifferent to each strategy that he or she chooses with a positive probability under his or her mixed strategy. Also, players’ randomizations are independent.8 A special case of a mixed strategy is a pure strategy, where probability 1 is placed on some action. It is easy to check that each mixing with probability 1/2 on L and R is the unique mixed strategy of the matching pennies game above. If a player, say the goalie, places weight of more than 1/2 on L, for instance, then the kicker would have a best response of choosing R with probability 1, but then that could not be an equilibrium as the goalie would want to change his or her action, and so forth. There is a deep and long-standing debate about how to interpret mixed strategies, and the extent to which people really randomize. Note that in the goalie and kicker game, what is important is that each player not know what the other player will do. For instance, it could be that the kicker decided before the game that if there was a penalty kick then she would kick to the left. What is important is that the kicker not be known to always kick to the left.9 We can begin to see how the equilibrium changes as we change the payoff structure. For example, suppose that the kicker is more skilled at kicking to the right side than to the left. 8An alternative definition of correlated equilibrium allows players to use correlated strategies but requires some correlating device that only reveals to each player his or her prescribed strategy and that these are best responses given the conditional distribution over other players’ strategies. 9The contest between pitchers and batters in baseball is quite similar. Pitchers make choices about the location, velocity, and type of pitch (e.g., whether various types of spin are put on the ball). If a batter knows what pitch to expect in a given circumstance, that can be a significant advantage. Teams scout one another’s players and note any tendencies or biases that they might have and then try to respond accordingly. 11 Electronic copy available at: https://ssrn.com/abstract=1968579 In particular, keep the payoffs as before, but now suppose that the kicker has an even chance of scoring when kicking right when the goalie goes right. This leads to the payoffs in Table 8. Table 8: A biased penalty-kick game Goalie L R Kicker L -1, 1 1, -1 R 1, -1 0, 0 What does the equilibrium look like? To calculate the equilibrium, it is enough to find a strategy for the goalie that makes the kicker indifferent, and a strategy for the kicker that makes the goalie indifferent.10 Let s1 be the kicker’s mixed strategy and s2 be the goalie’s mixed strategy. It must be that the kicker is indifferent. The kicker’s expected payoff from kicking L is −1 · s2(L) + 1 · s2(R) 11 and the payoff from R is 1 · s2(L) + 0 · s2(R), so that indifference requires that −s2(L) + s2(R) = s2(L), which implies that 2s2(L) = s2(R). Since these must sum to one (as they are probabilities), this implies that s2(L) = 1/3 and s2(R) = 2/3. Similar calculations based on the requirement that the goalie be indifferent lead to s1(L)− s1(R) = −s1(L), 10This reasoning is a bit subtle, as we are not directly choosing actions that maximize the goalie’s payoff and maximize the kicker’s payoff, but instead are looking for a mixture by one player that makes the other indifferent. This feature of mixed strategies takes a while to grasp, but experienced players seem to understand it well, as discussed below. 11To see where this payoff comes from, note that there is a s2(L) chance that the goalie also goes L and then the kicker loses and gets a payoff of -1, and a s2(R) chance that the goalie goes right and then the kicker wins and gets a payoff of 1; thus the expected payoff is −1 · s2(L) + 1 · s2(R) 12 Electronic copy available at: https://ssrn.com/abstract=1968579 that correspond to subsequent players who make choices. In Figure 1, player 1 has a choice of moving either left or right. The branches in the tree correspond to the different actions available to the player at a given node. In this game, if player 1 moves left, then player 2 moves next; while if player 1 moves right, then player 3 moves next. It is also possible to have trees in which player 1 chooses twice in a row, or no matter what choice a given player makes it is a certain player who follows, and so forth. The payoffs are given at the end nodes and are listed for the respective players. The top payoff is for player 1, the second for player 2, and the bottom for player 3. So the payoffs depend on the set of actions taken, which then determines a path through the tree. An equilibrium provides a prediction about how each player will move in each contingency and thus makes a prediction about which path will be taken; we refer to that prediction as the equilibrium path. We can apply the concept of a Nash equilibrium to such games, which here is a specifi- cation of what each player would do at each node with the requirement that each player’s strategy be a best response to the other players’ strategies. Nash equilibrium does not al- ways make sensible predictions when applied to the extensive form. For instance, reconsider the advertising example discussed above in Table 3. Suppose that firm 1 makes its decision of whether to advertise before firm 2 does, and that firm 2 learns firm 1’s choice before it chooses. This scenario is represented in the game tree pictured in Figure 2. To apply the Nash equilibrium concept to this extensive form game, we must specify what each player does at each node. There are two Nash equilibria of this game in pure strategies. The first is where firm 1 advertises, and firm 2 does not (and firm 2’s strategy conditional on firm 1 not advertising is to advertise). The other equilibrium corresponds to the one identified in the normal form: firm 1 does not advertise, and firm 2 advertises regardless of what firm 1 does. This is an equilibrium, since neither wants to change its behavior, given the other’s strategy. However, it is not really credible in the following sense: it involves firm 2 advertising even after it has seen that firm 1 has advertised, and even though this action is not in firm 2’s interest in that contingency. To capture the idea that each player’s strategy has to be credible, we can solve the game backward. That is, we can look at each decision node that has no successor, and start by making predictions at those nodes. Given those decisions, we can roll the game backward 15 Electronic copy available at: https://ssrn.com/abstract=1968579 Figure 2: Advertising Choices of Two Competitors and decide how player’s will act at next-to-last decision nodes, anticipating the actions at the last decision nodes, and then iterate. This is called backward induction. Consider the choice of firm 2, given that firm 1 has decided not to advertise. In this case, firm 2 will choose to advertise, since 13 is larger than 12. Next, consider the choice of firm 2, given that firm 1 has decided to advertise. In this case, firm 2 will choose not to advertise, since 7 is larger than 6. Now we can collapse the tree. Firm 1 will predict that if it does not advertise, then firm 2 will advertise, while if firm 1 advertises then firm 2 will not. Thus when making its choice, firm 1 anticipates a payoff of 7 if it chooses not to advertise and 13 if it chooses to advertise. Its optimal choice is to advertise. The backward induction prediction about the actions that will be taken is for firm 1 to advertise and firm 2 not to. Note that this prediction differs from that in the simultaneous move game we analyzed before. Firm 1 has gained a first-mover advantage in the sequential version. Not advertising is no longer a dominant strategy for firm 1, since firm 2’s decision depends on what firm 1 does. By committing to advertising, firm 1 forces firm 2 to choose not to advertise. Firm 1 is better off being able to commit to advertising in advance. A solution concept that capture found in this game and applies to more general classes of 16 Electronic copy available at: https://ssrn.com/abstract=1968579 games is known as subgame perfect equilibrium (due to Reinhard Selten (1975) Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games, International Journal of Game Theory 4:25 - 55). A subgame in terms of a finite game tree is simply the subtree that one obtains starting from some given node. Subgame perfection requires that the stated strategies constitute a Nash equilibrium in every subgame (including those with only one move left). So it requires that if we start at any node, then the strategy taken at that node must be optimal in response to the remaining specification of strategies. In the game between the two firms, it requires that firm 2 choose an optimal response in the subgame following a choice by firm 1 to advertise, and so it coincides with the backward induction solution for such a game. It is worth noting that moving first is not always advantageous. Sometimes it allows one to commit to strategies which would otherwise be untenable, which can be advantageous; but in other cases it may be that the information that the second mover gains from knowing which strategy the first mover has chosen is a more important consideration. For example, suppose that the matching pennies game we discussed above were to be played sequentially so that the kicker had to kick first and the goalie had time to see the kicker’s action and then to react and could jump left or right to match the kicker’s choice: the advantage would certainly then tip towards the goalie. This concludes our whirlwind tour of some of the basic tools of game theory. There are many important subjects that I have not touched upon here, including analyses that incorporate incomplete information, repeated games, and behavioral game theory. However, this should provide you with some feeling for a few of the most prominent concepts, and some of the approaches that form the backbone of game theoretic analyses. 3 Some Exercises Exercise 1 Product Choices. Two electronics firms are making product development decisions. Each firm is choosing between the development of two alternative computer chips. One system has higher efficiency, but will require a larger investment and will be more costly to produce. Based on estimates 17 Electronic copy available at: https://ssrn.com/abstract=1968579 Argue that there is no pure strategy Nash equilibrium to this game. Argue that mixing uniformly at random over all possible configurations of units is not a mixed strategy Nash equilibrium (hint - show that placing all units on one battlefield is an action that an army would not want to choose if the other army is mixing uniformly at random). Argue that each army mixing with equal probability between (0,3,3), (3,0,3) and (3,3,0) is not an equilibrium.14 Exercise 5 Divide and Choose. Two children must split a pie. They are gluttons and each prefers to eat as much of the pie as they can. The parent tells one child to cut the pie into two pieces and then allows the other child to choose which piece to eat. The first child can divide the pie into any multiple of tenths (for example, splitting it into pieces that are 1/10 and 9/10 of the pie, or 2/10 and 8/10, and so forth). Show that there is a unique backward induction solution to this game. Exercise 6 Information and Equilibrium. Each of two players receives an envelope containing money. The amount of money has been randomly selected to be between 1 and 1000 dollars (inclusive), with each dollar amount equally likely. The random amounts in the two envelopes are drawn independently. After looking in their own envelope, the players have a chance to trade envelopes. That is, they are simultaneously asked if they would like to trade. If they both say “yes,” then the envelopes are swapped and they each go home with the new envelope. If either player says “no,” then they each go home with their original envelope. The actions in this game are actually a full list of whether a player says yes or no for each possible amount of money he or she is initially given. To simplify things, let us write down actions in the following more limited form: an action is simply a number between 0 and 1000, meaning that if they get an envelope with more than that number, then they say “no” and otherwise they say “yes”. 14Finding equilibria to Colonel Blotto games is notoriously difficult. One exists for this particular version, but finding it will take you some time. 20 Electronic copy available at: https://ssrn.com/abstract=1968579 So, for instance, if player 1 chooses action “3”, then she says “yes” to a trade when her initial envelope has 1 or 2 or 3 dollars, but says “no” if her envelope contains 4 or more dollars. In a pure or mixed strategy equilibrium is it possible for both players to choose action “1000” with some positive probability? Suppose that player 2 does not play action “1000”, can a best response of player 1 involve any positive probability on the action “1000”? Repeat the above logic to argue that neither player will ever play “999” in an equilibrium. Iterating on this logic, what is the unique Nash equilibrium of this game? 21 Electronic copy available at: https://ssrn.com/abstract=1968579