Economics 335 March 2, 1999 Notes 6: Game Theory I. Introduction A. Idea of Game Theory Game theory analyzes interactions between rational, decision-making individuals who may not be able to predict fully the outcomes of their actions. B. Definition A game is a formal representation of a situation in which a number of decision makers (players) interact in a setting of strategic interdependence. By that, we mean that the welfare of each decision maker depends not only on her own actions, but also on the actions of the other players. Moreover, the actions that are best for her to take may depend on what she expects the other players to do. C. Elements of a Game 1. Players or decision makers a. firms b. consumers c. poker or chess players d. nature - nature chooses actions according to fixed probabilities 2. Rules of the game a. Who moves when? b. What do players know when they move (information structure)? c. What options are available to players at various points of the game? 3. Outcomes For each possible set of actions (strategies) by the players, what is the outcome of the game? 4. The payoffs What are the player s preferences over the outcomes? How do they rank these outcomes? D. Example 1 - Tick-Tack-Toe 1. Players There are two players denoted X and O. 2. Rules of the game a. The players are faced with a board that consists of nine squares arrayed with three rows of three squares stacked on each other as in Figure 1. b. The players take turns putting their symbol (X or 0) into an as yet unmarked square. c. Player X moves first. d. Both players observe all choices previously made.
2 3. Outcomes a. The first player to have three marks in a row (horizontally, vertically, or diagonally) wins and receives $1.00 dollar from the other player. b. If no one succeeds in winning after all nine boxes are marked, the game is a tie and no payments are made or received by either player. 4. Payoffs The payoff is the amount of money received and we assume that the players prefer more money to less money. Figure 1 E. Example 2 - Turkey sales 1. Players There are two players denoted Hy-Vee and Fareway 2. Rules of the game a. The game is played 7 days before the Thanksgiving Holiday. b. Both stores must deliver a weekly advertising supplement to the local paper stating the price per pound of frozen whole turkeys. c. The turkeys advertised are of the same quality. d. The players may only announce one of two prices : low and high. e. The stores do not know which price will be submitted by the other store, and the deadline for getting the advertisement in the paper is the same for both stores. 3. Outcomes The outcomes are specified using a matrix listing the net returns for each possible combination of choices by each firm where the first number in each cell represents the net returns to Fareway.
3 Table 1 Outcomes for Fareway and Hy-Vee from Advertising Decision Hy-Vee s Strategy Low Price High Price Low Price 1000 1000 1500 500 Fareway s Strategy High Price 500 1500 1200 1200 The idea is that if a store advertizes a high price and the other store advertizes a low price, the low price store will steal customers from the high price store. 4. Payoffs The payoff is the amount of money received and we assume that the players prefer more money to less money. II. Strategic or Normal Form Games A. Timing In a normal form game all players are assumed to make their moves at the same time. The pricing game above is in normal form while tick-tack-toe is not. B. Formal Description of the Game 1. A set of N players whose names are listed in the set I = {1,2,..., N). For the turkey example the set I is I = {1, 2} where 1 is Fareway and 2 is Hy-Vee. 2. For each player i, i I, there is a strategy set S i which contains all actions (or pure strategies) available to player i. We sometimes denote this set as the action set A i. A particular element of this set is denoted s i. At this point we assume that the strategy set is finite. The strategy set for each player in the turkey pricing game is S i = {low, high}. 3. An outcome of the game is a listing of the strategies chosen by each player and is denoted by s = (s 1, s 2,...,s i,... s N-1, s N ). The possible outcomes in the turkey pricing game are (low, low), (low, high), (high, low) and (high, high). We denote the set of all actions except that of ith player as s i = (s 1, s 2,...,s i-1, s i+1,... s N-1, s N ). 4. Each player has a payoff function u i (s 1, s 2,...,s i,... s N-1, s N ) which assigns a real number to each outcome of the game. Formally, each payoff function u i maps the N-dimensional vector, s, into the real line. It is assumed that players prefer higher numbers to lower ones. The payoff functions in the turkey pricing game are as follows: u 1 ( low,low) = 1000 u 2 (low,low) = 1000 u 1 ( low, high) = 1500 u 2 (low,high) = 500 u 1 ( high,low) = 500 u 2 (high,low) = 1000 u 1 ( high, high) = 1200 u 2 (high,high) = 1200 It is clear in this game that total net returns are highest if both firms choose high prices while total returns are the same for all other choices.
4 C. Equilibrium of the Game using Dominant Strategies 1. Definition of dominant strategies A particular strategy s i players choose, playing s i always maximizes player i s payoff. Formally, for every choice of all strategies for all players except i, s i, g S i is said to be a dominant strategy for player i if no matter what all other u i ( s i,s i )>u i (s i,s i ), for every s i g S i We say that a strategy is a dominated strategy for a player if there exists another strategy which is always strictly better, no matter what strategies are chosen by the other players. A weakly dominant strategy for a player in one that is weakly better (as good as or better) than every other strategy for that player, no matter what strategies are chosen by the other players. A strategy is weakly dominated for a player if there exists another strategy which is always weakly better, no matter what strategies are chosen by the other players. 2. Example Consider Fareway in the turkey pricing game. If Hy-Vee chooses a low price then Fareway will get 1000 if it chooses a low price and 500 if it chooses a high price so Fareway will choose a low price if it knows Hy-Vee has chosen a low price. If Hy-Vee chooses a high price, Fareway will get 1500 if it chooses a low price and 1200 if it chooses a high price so Fareway will choose a low price if it knows Hy-Vee will choose a high price. Therefore the optimal strategy for Fareway in either case is a low price. Therefore we say that a low price is a dominant strategy for Fareway. If we were to perform the same exercise for Hy-Vee we would find that Hy-Vee also has a dominant strategy of choosing the low price. 3. Definition of equilibrium in dominant strategies An outcome ( s 1, s 2,..., s N ) where s i g S i for every i ' 1,2,...,N equilibrium in dominant strategies if s i is a dominant strategy for each player i. is said to be an 4. Example of dominant strategy equilibrium In the above turkey pricing game, the outcome (low, low) is a dominant strategy equilibrium since low is a dominant strategy for both Hy-Vee and Fareway. This result is somewhat surprising since if both stores were to advertise the high price, they would be better off. The point (high, high) is called Pareto superior to the point (low, low) since both firms are better off at (high, high).
5 5. Situations with no dominant strategy Consider a different pricing game, this time between two local agricultural cooperatives who are placing advertisements for tires in the local newspaper. The tires are the same brand (Unmarked and Bland Incorporated) and style. Let the two firms be denoted FCS (Farm Coop Society) and LCI (Local Coop Incorporated). Assume that there are 3 elements of the strategy set (low, medium, high) for each cooperative. The outcome matrix for this problem is as follows. Table 2 Outcomes for FCS and LCI from Advertising Decision LCI s Strategy Low Price Medium Price High Price Low Price 1000 1000 1600 1200 2000 1500 FCS s Strategy Medium Price 1200 1600 2000 2000 2500 1800 High Price 1500 2000 1800 2500 2200 2200 The idea is that a store s returns will go up as the other store raises its price. However, holding the other store s price constant, a store s revenue may go up as it raises its own price if sales do not fall too much. For example when LCI s price is low, FCS does better by raising its price from low to medium and from medium to high. When LCI s price is medium, FCS does better by raising its price from low to medium, but does worse by raising it from medium to high, since sales fall more than is compensated for by the higher price. Consider now the optimal strategies for FCS. If LCI chooses a low price, FCS prefers a high price. If LCI chooses medium, FCS does best with a medium price, while if LCI chooses high, FCS also does best with a medium price strategy. Thus there is no dominant strategy for FCS since sometimes it prefers a medium price and sometimes it prefers a high price. We can say, however, that the low price strategy is dominated since it is never optimal. If we perform the same exercise for LCI, we find that when FCS chooses a low price, LCI does best with a high price, while when FCS chooses either medium or high, LCI does best with a medium price. 6. Iterated dominant strategy equilibrium In the tire pricing game, there is no dominant strategy equilibrium since at least one player does not have a dominant strategy. But this game still has an equilibrium that can be found by iterating the decision process. Since FCS knows that it will never choose a low price, we can eliminate the first row of Table 2. And since LCI will never choose a low price, we can also eliminate the first column. The information is reproduced in Table 3. Table 3 Outcomes for FCS and LCI from Advertising Decision with Low Prices Excluded ICI s Strategy Medium Price High Price FCS s Strategy Medium Price 2000 2000 2500 1800 High Price 1800 2500 2200 2200
6 Now consider the high price strategy for FCS. It is dominated by the medium price strategy whether LCI chooses medium or high so the medium price strategy is dominant for FCS. We can also show that the medium price strategy is dominant for LCI. Thus the equilibrium of this game and the one in Table 2 is (medium, medium). D. Equilibrium of a Game Using the Concept of Nash Equilibrium 1. Games with no equilibrium in dominant strategies Many games have no equilibrium in dominant strategies, particularly if one player does not have a dominant strategy. In many cases, there is not an iterated equilibrium either. 2. Example - Standards for power takeoff (PTO) connections Consider a game played by John Deere and New Holland. Assume that John Deere is considering a new standard for power takeoff connections for its next model year tractors. We will label the new standard as new and the current one as old in the following matrix representation of the game. While not totally realistic we will assume that this is a game of simultaneous moves in that the dealers must come to market before knowing which standard the other has adopted. Table 4 Outcomes for John Deere and New Holland Adoption Decision on PTO standards John Deere s Strategy Old Standard New Standard New Holland s Strategy Old Standard 200 1000 100 800 New Standard 100 800 300 1200 First consider if there is an equilibrium in dominant strategies. Since New Holland prefers old when John Deere chooses old and new when John Deere chooses new, there is no equilibrium in dominant strategies. Since there are only two choices, the problem has no meaning if we eliminate one of the strategies. Thus there is no iterated equilibrium in dominant strategies. Given what we know so far, we cannot predict an equilibrium for this game. 3. Definition of Nash equilibrium The most common equilibrium concept used in game theory is due to John Nash and is called Nash equilibrium. The idea is that an outcome is an equilibrium if no player would find it to her advantage to deviate from this point. Specifically: s i g S i for every i ' 1,2,...,N. s ' ( s 1, s 2,..., s N ) Suppose that Then the strategy profile is said to be a Nash equilibrium (NE) if no player would benefit by deviating, provided that no other player deviates either. Formally, for every player 1, i = 1, 2,..., N, u i ( s i, s i ) $ u i (s i, s i ), for every s i g S i Notice the difference between this and a dominant strategy equilibrium is that the second argument in. The strategy need not be better in all circumstances u i (@, @) is s i and not s i s i (against all possible strategies s i ), but merely better against the specific strategy profile s i, which is played in the equilibrium.
7 4. Example Now investigate the PTO game and Nash equilibrium. Consider the outcome (old, old). If John Deere chooses old then the best strategy for New Holland is to choose old. If New Holland chooses old, the best strategy for John Deere is to choose old. Thus the outcome (old, old) is a Nash equilibrium since neither player has an incentive to deviate from it. The same is true of the (new, new) strategy. The strategy (old, new) is not a Nash equilibrium since if John Deere chooses new, New Holland will want to choose old. Similarly for the outcome (new, old). Thus this game has 2 Nash equilibriums and we cannot really predict which will be chosen. 5. Nash equilibrium and dominant strategy equilibrium It can be easily shown that any dominant strategy equilibrium is a Nash equilibrium, but not vice versa. E. Non-existence of Nash Equilibrium 1. Definition Many games have no equilibrium in dominant or iterated dominant strategies and may not even have a Nash equilibrium in pure strategies. (Players do not randomize their choice of actions as, for example, a baseball pitcher does when choosing which sort of ball to throw.) A game has no equilibrium in pure strategies if at every possible outcome, at least one player has an incentive to deviate. 2. Example Consider the following game between Farmer Slack and Freddy Foreclose. The game involves the two players deciding where to spend the morning. They both leave their homes at the same time. Freddy s payoffs are higher if he ends up at the same place as Farmer Slack since he can then deliver the foreclosure papers. Farmer Slack is better off if she can avoid Freddy. Both of them prefer the Coffee Shop to the Corn Silage Pit. Consider then the matrix of payoffs for this (cat and mouse) game. Table 5 Outcomes for Farmer Slack and Freddy Foreclose from the decision of where to spend the morning. Farmer Slack Silage Pit Coffee Shop Freddy Foreclose Silage Pit 400 0 0 500 Coffee Shop 100 200 1000 50 There are four possible outcomes of this game. Consider each of them in turn. a. (Silage Pit, Silage Pit) For this outcome u Slack (Silage Pit, Coffee Shop) = 500 > 0 = u Slack (Silage Pit, Silage Pit) so if Freddy plays silage pit, Slack will play Coffee Shop. Thus this is not a Nash equilibrium. b. (Silage Pit, Coffee Shop) For this outcome u Freddy (Coffee Shop, Coffee Shop) = 1000 > 0 = u Freddy (Silage Pit, Coffee Shop) so if Slack plays coffee shop, Freddy will play Coffee Shop. Thus (Silage Pit, Coffee Shop) is not an equilibrium.
8 c. (Coffee Shop, Silage Pitt) For this outcome u Freddy (Silage Pit, Silage Pit) = 400 > 100 = u Freddy (Coffee Shop, Silage Pit) so if Slack plays Silage Pit, Freddy will play Silage Pit. Thus this is not an equilibrium. d. (Coffee Shop, Coffee Shop) For this outcome u Slack (Coffee Shop, Silage Pit) = 200 > 50 = u Freddy (Silage Pit, Coffee Shop) so if Freddy plays coffee shop, Slack will Silage Pit. Thus (Coffee Shop, Coffee Shop) is not an equilibrium. This game then has no Nash equilibrium outcome. F. Mixed Strategy Equilibrium Although this game does not have an equilibrium in pure strategies, if we expand our concept of a strategy, it does have an equilibrium. The reason why we could find no equilibrium before is that, whatever Farmer Slack does, Freddy Foreclose will want to be in the same place as Slack, but then Slack will want to move to the other place. As long as Freddy knows where Slack is, Slack gets a bad payoff. Therefore, what Slack really must do, in order to do well, is to be unpredictable. She must randomize her choice to keep Freddy guessing. We call such a strategy a mixed strategy. Freddy will also want to play a mixed strategy. As long as Slack knows where Freddy is, she will choose to go elsewhere. To have any chance of finding Slack, Freddy must make sure that Slack does not know exactly where he will be. In general, as in this game, when one player has a mixed strategy in the equilibrium, so will the other. To have mixed strategy equilibrium, it must be the case that the players are indifferent between the actions which they choose to play. If a player does strictly better with one action or another, then he will not want to play a mixed strategy. Thus we solve for the mixed strategy equilibrium by finding the randomization choice which makes the other player indifferent between actions. In this case, suppose that Farmer Slack chooses the silage pit with probability p, and chooses the coffee shop with probability 1-p. Then by choosing the silage pit, Freddy gets the payoff 400p + 0(1-p) = 400p, and by choosing the coffee shop, Freddy gets 100p + 1000(1-p). For Freddy to be indifferent between the silage pit and the coffee shop given Farmer Slack s choice of p, it must be that p satisfies 400p ' 100p % 1000(1&p) Y p ' 10 13 Thus, to keep Freddy guessing, Slack must go to the silage pit 10/13 of the time, and go to the coffee shop 3/13 of the time. If Farmer Slack goes to either place more often than this, then Freddy will prefer one action over the other, and it will not be an equilibrium. (Since if Freddy does one action all the time, Slack will not want to randomize.) We can see this another way by considering the best strategy for Farmer Slack given her belief s about what Freddy thinks she will do. Suppose Farmer Slack thinks that Freddy thinks that she will choose the silage pit with probability.60 Then she can compute Freddy s expected payoffs as follows (assuming he chooses the silage pit with probability q). E( Freddy p'.60) '.6q(400)%.6(1&q)(100)%.4q(0)%.4(1&q)(1000) ' 240q%60&60q% 0%400&400q ' 460&220q Freddy will maximize his payoffs by setting q equal to 0 and have expected payoff equal to 460. Thus even though Freddy knows Slack will be in the Silage Pit 60% of the time, it is such an undesirable place he would
9 rather just stay at the coffee shop and have an expected payoff of 460 (400 from the 40% of the time Slack comes and 60 from the 60% of the time she doesn t). This is not a random strategy and so her expectation that Freddy will play a random strategy is wrong. Given that Slack has figured out that Freddy will just stay in the coffee shop if she plays silage pit with probability.6, she will now not play a random strategy but play silage pit all the time. But now Freddy will play silage pit and we are back to where we started. Now suppose Slack thinks that Freddy thinks that she will randomize with probability of.80 on the silage pit. Freddy s expected profits are now as follows. E( Freddy p'.80) '.8q(400)%.8(1&q)(100)%.2q(0)%.2(1&q)(1000) ' 320q%80&80q% 0%200&200q ' 280%40q Freddy now does best by making q as large as possible or choosing the silage pit with probability 1. This gives an expected payoff of 320 which is less than last time but more than if he chose the coffee shop with any positive probability. Even though Freddy hates the silage pit he will stay there all the time for the 80% chance of getting Slack. Again Freddy does not randomize but now chooses the silage pit with certainty. Only when p = 10/13 will Freddy not choose q equal to zero or one. Specifically E( Freddy p' 10 13 ) ' 10 10 q(400)% 13 13 (1&q)(100)% 3 13.2q(0)% 3 13 (1&q)(1000) ' 4000 1000 q% 13 13 & 1000 3000 q% 0% 13 13 & 3000 13 q ' 4000 13 ' 307.692 In this case any value of q will give the same expected payoff and so Freddy can randomize with no change in the expected return. We can also calculate what Freddy s strategy must be. Suppose that Freddy goes to silage pit with probability q. Then, by going to the silage pit, Slack gets a payoff of (0)q + 200(1-q), and by going to the coffee shop, she gets 500q + 50(1-q). For Slack to be indifferent or to sometimes do one and sometimes do the other, we must have 200(1&q) ' 500q % 500(1&q) Y q ' 3 13 In the equilibrium, Freddy spends 3/13 of the time at the silage pit, and spends 10/13 of the time at the coffee shop. What is the probability that Farmer Slack will be caught by Freddy Foreclose? Example: Another real-world example of a mixed strategy equilibrium is in the game of baseball. The pitcher always wants to keep the batter guessing about what kind of ball she will throw. If the batter knows which type of ball to expect, he can tailor his response to this kind of ball and hit a home-run. By randomizing, the pitcher forces the batter to choose some sort of generic response, or forces the batter to randomize between different types of responses, in the hope that at least some of the time his response will be appropriate and he will be lucky enough to hit a home run anyway.
10 G. The Path to Nash Equilibrium In the games we have to this point we assume that there is perfect information in that all the players know the payoffs of all other players. Given this fact we can sometimes predict the equilibrium of the game by considering alternative hypothetical beliefs for each player and checking whether they lead to a Nash equilibrium. Consider the following game between two competing feed companies who are planning to run a sale on a particular premix. Assume that the two companies are Clark s Feeds and Fenman s Feeds. The payoffs to alternative strategies are given in Table 6 where the first element of a box is the payoff to Clark. Clark has a reputation of sometimes knocking a little off the sale price (with sufficient negotiating and small talk) while Fenman has never been known to give a price break to anyone. Given this fact, Clark will always do better by advertising the same price as Fenman, since some customers will come because they know they can beat the price advertised little bit by negotiating. On the other hand, Fenman often does best by not advertising the same price as Clark. For example, Fenman does best by advertising low when Clark advertises low, and does best by advertising medium when Clark advertises high. However, it turns out to be better to advertise high when Clark advertises medium since some customers don t like haggling over small price breaks, and besides, Clark has a reputation for occasionally watering the feed. Consider the table that follows. Table 6 Outcomes for Clark and Fenman from Advertising Decision Fenman s Strategy Low Price Medium Price High Price Low Price 100 200 150 150 200 100 Clark s Strategy Medium Price 80 220 200 225 250 230 High Price 75 250 150 260 280 250 There is no equilibrium in dominant strategies since when Clark picks low, Fenman will pick low, but when Clark picks medium, Fenman will pick high. Thus Fenman has no dominant strategy. The same is true for Clark. You can verify that there is no iterated dominant strategy equilibrium either. Now consider how to find a Nash equilibrium sequentially. Because Clark is rational, he will choose the strategy that is best, given some belief about the strategy that Fenman will choose. And because Clark believes that Fenman is also rational, he knows that Fenman will choose the best strategy given some belief about what Clark will do. So start by supposing (for the sake of argument) that Clark believes that Fenman thinks that Clark will advertise a high price. In this case, Clark believes that Fenman will advertise a medium price, since 260 is the largest payoff for Fenman when Clark advertises a high price. But if Fenman advertise a medium price, then Clark does best by advertising a medium price, not a high price. Thus Fenmans hypothesis turns out to be wrong. Clarks conjecture about what Fenman believes about him is inconsistent with optimal behavior for each player and thus is not rational. The result is similar if Clark believes that Fenman thinks that Clark will advertise a medium price. If Fenman thinks Clark will advertise a medium price, then Fenman will advertise a high price. But if Fenman advertises a high price, Clark does best by advertising a high price, not a medium price, since 280 is the largest payoff for Clark when Fenman chooses a high price. Thus Clark s conjecture is not consistent with rational behavior on the part of each player.
11 But now suppose that Clark believes that Fenman thinks that Clark will advertise a low price. In this case if Fenman thinks Clark will advertise a low price, then Fenman will advertise a low price as well. And if Fenman advertises a low price, Clark does best by advertising a low price, since 100 is the largest payoff for Clark when Fenman advertises a low price. Thus Clark does exactly what Fenman expects Clark to do. Clark s conjecture is consistent with rational behavior on the part of each player. We have an equilibrium. We can summarize this process for Clark in a table: Table 7 Clark s best strategy for each belief he could have about Fenman Clark s belief about Fenman s belief about his strategy Clark s belief about Fenman s optimal strategy Clark s best strategy Low Price Low Price Low Price Medium Price High Price High Price High Price Medium Price Medium Price We can construct a similar table for Fenman summarizing her best strategy considering what she believes Clark thinks that she will do. Table 8 Fenman s best strategy for each belief she could have about Clark Fenman s belief about Clark s belief about her strategy Fenman s belief about Clark s optimal strategy Fenman s best strategy Low Price Low Price Low Price Medium Price Medium Price High Price High Price High Price Medium Price The point is that the set of beliefs and actions (low, low) are mutually consistent for both players. This is a Nash equilibrium, as you can verify. H. Best Response Functions 1. Definition A best response function for a player in an N-person game gives the optimal strategy for a given player given the strategies of other players. Specifically, the best response function for player i is the function R i (s i ), which for given strategies s i of players 1,2,..., i-1, i+1,...,n, assigns a strategy s i = R i (s i ) that maximizes player i s payoff or utility u i (s i, s i ). 2. Proposition concerning Nash equilibrium If s is a Nash equilibrium outcome, then s i ' R i ( s i ) for every player i.
12 3. Example for the feed pricing game The best response function for Clarks is as follows: R clarks s Fenmans ' low if s Fenmans ' low medium if s Fenmans ' medium high if s Fenmans ' high The best response function for Fenmans is as follows: R Fenamns s clarks ' low if s Clarks ' low medium if s Clarks ' high high if s Clarks ' medium We can find a Nash equilibrium, if it exists, by finding points that are on the best response functions of both players. In this game there are nine possible outcomes. Consider the outcome (low, medium). For Clarks, R Clarks (medium) = medium which is not low so this is not an equilibrium. Now consider the outcome (high, medium). For Fenmans, R Fenmans (high) = medium so this is a best response for Fenmans. But consider whether this is a best response for Clarks. R Clarks (medium) = medium which is not high, so this is not an equilibrium. But consider (low, low). Here R Clarks (low) = low and R Fenmans (low) = low and this point is a Nash equilibrium.
13 III. Extensive Form Games A. Idea Extensive form games are ones in which players make moves in a sequence and may move more than once. Some are finite and some infinite in length. Normal form games played several times (repeated) are a type of extensive form game. B. Review of the Elements of a Game 1. Players or decision makers a. decision-making entities b. nature - nature chooses actions according to fixed probabilities 2. Rules of the game a. Who moves when or the order of play? 1) simultaneous move games 2) sequential move games b. What options are available to players at various points of the game? 1) Decision nodes are points of the game where an individual player is called upon to make a decision. 2) The location of the nodes and the options available at the node are both important c. What do players know when they move (information structure)? 1) A game has perfect recall if no player forgets any information she once knew, and each players know the actions he or she has taken. 2) If every player at every decision node knows the actions taken previously by every other player (including nature), then the game is one of perfect information. Because games of imperfect information are much more difficult to solve, we will study only games of perfect information here. 3) We also assume that all players are rational and this fact is known by all players. 3. Representation of rules by a game tree a. Nodes are decision points. b. Branches represent choices that are open to the decision maker. c. Every branch connects two nodes, one of which is a parent and one of which is a child. d. Every node has at most one parent and is connected to the parent by one branch. e. If a node B comes after node A then node A is called an ancestor of B and B is called a successor of A.
14 f. A node with no ancestors is called an initial node. g. A node with no successors is called a terminal node. All other nodes (including initial nodes) are called decision nodes. h. Every game tree has one initial (starting) node. All other nodes are descendants of this one. i. example game diagram Consider a game with two players, an incumbent in the industry and a potential entrant. The incumbent moves first and chooses a technology. The potential entrant then decides whether to enter or not. Based on the technology chosen by the incumbent and the entry decision, the firms receive different pay-offs. The initial node is I1. The terminal nodes are T1, T2, T3, T4. Node E1 is a parent and ancestor of T2. Node I1 is a parent of E2 and an ancestor of T4. The branches emanating from I1 are Technology A and Technology B while the branches emanating from E1 are Enter and Do not enter. Extensive Form Game Tree Entry Deterrence I E1 Enter T1 (1000, -100) Entrant Technology A Do not enter T2 (2000, 0) Incumbent I1 Enter T3 (500, 500) Technology B Entrant E2 Do not enter T4 (3000, 0) 4. Outcomes For each possible set of actions (strategies) by the players, what is the outcome of the game? The outcomes in the diagram above are represented by the terminal nodes (T1-T4) which are determined by the strategies chosen by the players.
15 5. The payoffs What are the player s preferences over the outcomes? How do they rank these outcomes? The payoffs in the above game are given by the numbers next to the terminal nodes. The first number in each pair is the payoff to the incumbent at that terminal node. Players prefer more to less. C. Formal Definition of an Extensive Form Game 1. A game tree containing a starting node, other decision nodes, terminal nodes, and branches linking each decision node to a successor node. 2. A list of N $ 1 players, indexed by i, i = 1, 2, 3,..., N. 3. For each decision node, the name of the player entitled to choose an action. 4. For each player i, a specification of i s strategy (action ) set at each node that player i is entitled to choose an action. 5. A specification of the payoff to each player at each terminal node. D. Strategies for Extensive Form Games 1. Definition In a normal form game, a strategy and an action are synonymous since actions are simultaneous and played only once. In an extensive form game, a player must consider what actions she would take under different circumstances, after different choices of the other players who move earlier in the game. A strategy for player i (denoted s i ) is a complete plan (list) of actions, one action for each decision node at which the player is entitled to choose an action, corresponding to a set of decisions by other players who move earlier. 2. Example for the entry game Consider the strategies available to the entrant. Since the entrant may end up at either E1 or E2, a strategy is a specification of the precise actions to be taken at both nodes. While it is clear that the entrant will end up at either E1 or E2 and not both, a strategy must specify an action at each of the two nodes. Therefore the entrant has 4 possible strategies in this case. They are denoted (Enter, Enter), (Enter, Don t Enter), (Don t Enter, Enter), (Don t Enter, Don t Enter) where the first element of the pair denotes the action at node E1. For example, the first strategy says to enter regardless of the technology chosen by the incumbent. The strategy set for the incumbent consists of the action set at node I1 or (A, B). 3. Writing an extensive form game in normal (strategic) form Once we specify the complete strategy set for all players, we can represent an extensive form game in normal form by assuming the all players must announce their strategies at the same time. In the entry game, recall that I has two strategies and E has four strategies. We can express this game in a table, just like any normal form game:
16 Table 8 Outcomes for Entry Game I in Normal Form Entrant s Strategy (E, E) (E, D) (D, E) (D, D) Technology A (1000, -100) (1000, -100) (2000, 0) (2000, 0) Incumbent s Strategy Technology B (500, 500) (3000, 0) (500, 500) (3000, 0) Looked at in normal form, there are eight possible outcomes: (A, (Enter, Enter)), (A, (Enter, Don t Enter)), (A, (Don t Enter, Enter)), (A, (Don t Enter, Don t Enter)) (B, (Enter, Enter)), (B, (Enter, Don t Enter)), (B, (Don t Enter, Enter)), (B, (Don t Enter, Don t Enter)). Of course, some of these outcomes lead to the same final nodes in the extensive form game tree. For example, (A, (Enter, Enter)) and (A, (Enter, Don t Enter)) lead to the same final node. But from the strategic perspective, these outcomes are different because the strategies which produce them are different. E. Equilibria for Extensive From Games 1. Nash equilibria for extensive form games written in normal form Since we have reduced the extensive form game to a normal form game, we can try to find the Nash equilibria for the game and see if they provide a sensible solution. For the example game, the point (A, (D, E) is a Nash equilibria. If the incumbent chooses technology A, then the entrant is indifferent between (D, E) and (D, D) but prefers these to other points. If the entrant chooses (D, E) then the incumbent prefers technology A. This is the only Nash equilibrium in the normal form game and represents an equilibrium of the extensive form game. 2. Multiple Nash equilibria for extensive form games written in normal form. Many games written in normal form have more than one Nash equilibrium. If the original game is in normal form then it may be difficult to choose between these equilibria. In many cases, in if the underlying game is one in extensive form, in which there is a sequential order of play, then we may be able to select one of these equilibria as being more likely. Consider the following simple example where the incumbent has a monopoly position in the market. Given this position and no rivals, the incumbent makes $100.00. The entrant obviously makes zero in this case. If the potential rival enters the market, the incumbent can follow one of two strategies: accommodate or fight. If the incumbent engages in a costly price war its returns decrease to $30.00 and the entrant loses $15.00. If it accommodates the rival and acts as a duopolist, the returns to the incumbent are $60.00 while the entrant will make $30.00. This can be modeled as a two stage game in which the entrant moves first and the incumbent then chooses to accommodate or fight. The game tree in extensive form is as follows.
17 Game Tree Entry Deterrence II I1 Fight T1 (-15,30) Incumbent Enter Accomodate T2 (30, 60) Entrant E1 Stay Out T3 (0, 100) As before we can also write this game in normal form. The payoff matrix is as below. Table 9 Outcomes for Entry Game II in Normal Form Incumbent s Fight Accommodate Enter (-15, 30) (30, 60) Entrant s Strategy Stay Out (0, 100) (0, 100) We can find the Nash equilibria as usual. There are two Nash equilibria to this normal form game (Enter, Accommodate) and (Stay Out, Fight). This is clear since if the entrant chooses to enter, the best strategy the incumbent is to accommodate, while if the incumbent is going to accommodate, the best strategy for the rival is to enter. Similarly if the entrant chooses to stay out, the incumbent may as well fight as accommodate, while if the incumbent is going to fight, the incumbent is better off to stay out. Thus this game seems to have two Nash equilibria. However, the outcome (Stay Out, Fight) is not really a sensible prediction for the game, since the entrant can foresee that if it does enter, the incumbent will in fact accommodate. Thus the concept of Nash equilibrium seems to be lacking in this game.
18 3. Threats (credible and incredible) a. When one player attempts to get other players to believe it will employ a specific strategy, it is called a threat. In the entry deterrence game II, the incumbent would like the entrant to believe that it will fight if the entrant chooses to get in the market. b. A threat by a player is not credible unless it is in the player s own interest to carry out the threat when given the option. c. Threats that are not credible are termed incredible. In the entry deterrence II game, the threat to fight if the rival enters the market is incredible. d. We may be able to eliminate some Nash equilibria as being unreasonable predictions for a game if we can show they depend on incredible threats. 4. Subgames Consider as an example the game played between a couple, John and Mary. They are planning to go out on a Friday evening and John needs to make reservations at a restaurant since Mary will not have access to a telephone on her job as a wildlife biologist. John prefers the local Sport s Grill while Mary prefers the new Thai restaurant. Before leaving for work they engage in a somewhat heated discussion and Mary says, If you make the reservation at the Sport s Grill, I ll just stay home with the four young children and watch cartoons and you can go out yourself. Later that day at work, John must decide which restaurant to call. Written in normal form this game would have 2 Nash equilibria where John and Mary both end up at one restaurant or the other since neither prefers Friday night at home. However, we can predict the outcome in the extensive form, since Mary s threat to just stay home is not credible when John gets to move first. Once a reservation has been made, Mary prefers to go the Sport s Grille as opposed to staying home. Knowing this, John will not make a reservation at the Thai restaurant (assuming that getting Mary upset doesn t affect his payoff, in which case he may choose differently). A subgame is a decision node from the original game along with the decision nodes and terminal nodes directly following this node. A subgame is called a proper subgame if it differs from the original game. Entry deterrence I has three subgames: the game itself and two proper subgames beginning with nodes E1 and E2. Entry deterrence II has two subgames: the game itself and the proper subgame with initial node I1. Just as games may have Nash equilibria, subgames may have Nash equilibria.
19 5. Using subgames to refine Nash equilibria in extensive form games a. example 1 To see how the idea of a subgame can help us predict the outcome of an extensive form game, consider entry deterrence II. Looking at the game tree, the optimal strategy for the incumbent in the subgame is to accommodate if the potential rival enters. Thus we can eliminate the choice (fight) from this decision node. The reduced game tree looks as follows: Pruned Game Tree Entry Deterrence II I1 Incumbent Enter Accomodate (30, 60) T2 Entrant E1 Stay Out T3 (0, 100) Having eliminated the fight option at node I1, it is clear that the best strategy for the entrant is to enter the market since 30 is greater than 0. The incumbent can threaten to fight if the rival enters but this threat is not credible. b. example 2 Consider now a new competition game which we denote entry deterrence III. In this game the there is one firm in the industry and one potential rival which is considering production. The firms use the same production technology and thus have the same cost of production per unit which we assume is constant. The larger firm is considering building a larger plant that has high fixed costs but will have lower and constant marginal costs than the current plants. This technology is really open only to the incumbent in the industry since the smaller firm does not have the distribution network to effectively market the additional product. Once the large firm decides whether to invest in the new plant or not, the rival decides whether to enter and each firm chooses an output level. The game tree is below.
20 Extensive Form Game Tree Entry Deterrence III E1 Enter T1 (4000, -2000) Entrant New Plant Do not enter T2 (5000, 0) Incumbent I1 Enter T3 (4500, 1500) Older Plant Entrant E2 Do not enter T4 (6000, 0) We can also represent this game in normal form with a payoff matrix as follows. Table 10 Outcomes for Entry Game III in Normal Form Entrant s Strategy (E, E) (E, D) (D, E) (D, D) New Plant (4000, -2000) (4000, -2000) (5000, 0) (5000, 0) Incumbent s Strategy Old Plant (4500, 1500) (6000, 0) (4500, 500) (6000, 0) We can find two Nash equilibria to this game. The first is (New Plant, (D, E)) since if the incumbent picks a new plant the best option for the rival is to stay out. But if the rival picks (D, E) the best choice for the incumbent is the new plant. The second is (Old Plant, (E, E)) since if the incumbent chooses an old plant the best strategy for the rival is to enter. But if the rival chooses to enter, the best strategy for the incumbent is to choose the old plant. The normal form game obscures the fact that the rival can observe whether the incumbent is building the new plant.
21 5. Subgame-perfect equilibrium As an alternative to Nash equilibrium in extensive form games, a refinement know as subgame perfect equilibrium is used. An outcome is said to be a subgame perfect equilibrium (SPE) if it induces a Nash equilibrium in every subgame of the original game (including the game itself). Consider now the two Nash equilibria in the entry deterrence III game. The first equilibrium is (New Plant, (D, E)). In the subgame starting at E1, the action D is optimal for the entrant since 0 > -2000. So the action D is a Nash equilibrium for this simple sub-game. Similarly if we consider the subgame starting at E2, the action E is optimal for the entrant since 1500 > 0. Thus the strategy set of the entrant (D, E) is optimal for both subgames of the original game and constitutes a subgame-perfect equilibrium. Now consider the second equilibrium of (Old Plant, (E, E)). In the subgame starting at E1, the action D is optimal for the entrant since 0 > -2000. So the action E is not a Nash equilibrium for this simple subgame and the strategy (E, E) cannot be part of a subgame-perfect equilibrium of the original game. Therefore, (Old Plant, (E,E)) is not subgame-perfect.. Every subgame-perfect equilibrium is a Nash equilibrium but not every Nash equilibrium is a subgameperfect equilibrium. 6. Backward induction a. algorithm In extensive games with perfect information, we can find the subgame perfect equilibrium using the following technique called backward induction. 1) Start at the terminal nodes of the game tree and trace each back to its parent. Each of these parent nodes is a decision node for some player. 2) Find the optimal decision for that player at that decision node by comparing the ranking the player assigns to the terminal nodes that are reached from this decision node. Record this choice at the node. 3) Prune from the tree all branches that originate from the decision nodes selected in step 1. The pruning now makes each of these decision nodes into a terminal node. Attach to each of these new terminal nodes the payoff received when the optimal action is taken at each node. 4) If this new game tree has no decision nodes, quit. 5) If the new tree still has decision nodes, return to step 1 and continue. 6) For each player collect the optimal decisions at every decision node that belongs to that player. This collection of decision constitutes the player s optimal strategy in the game.
22 b. example For entry deterrence III consider the following set of decision trees. Pruned Game Tree 1 Entry Deterrence III E1 Enter T1 (4000, -2000) Entrant New Plant Do not enter T2 (5000, 0) Incumbent I1 Enter T3 (4500, 1500) Older Plant Entrant E2 Do not enter T4 (6000, 0) Pruned Game Tree 2 Entry Deterrence III E1 Do not enter Entrant (5000, 0) New Plant Incumbent I1 Older Plant Entrant E2 Enter (4500, 1500) Pruned Game Tree 3 Entry Deterrence III New Plant Incumbent (5000, 0) I1
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