January 26, 2015 Exercise 9 7.c.1, 7.d.1, 7.d.2, 8.b.1, 8.b.2, 8.b.3, 8.b.4,8.b.5, 8.d.1, 8.d.2 Example 10 There are two divisions of a firm (1 and 2) that would benefit from a research project conducted by the headquarters ()ofthefirm. The value of the project to division is and the division receives 0 if the project is not provided. Providing the project would cost an amount. is concerned about the welfare of the entire firm. It therefore would like to provide the project if and only if 1 + 2 Each division knows the value that it would receive from the project if it is provided., however, does not know these values and must ask the divisions to report them. If a division anticipates how its report will influence the outcome, then it may choose to misrepresent the benefit that it would receive from the project. In the discussion below, let denote a reported benefit by division. Suppose decides to charge division the amount = when 1 + 2. Here, again denotes the report of the other division. No money is exchanged when the project is not provided. This may not seem particularly intuitive as a rule for the price, but it has several virtues that will be made apparent through this example. 1. What is division s payoff as a function of its report and the report of the other division? ½ ( ) + = if + 0 otherwise 2. Show that reporting honestly (i.e., = ) dominates any report that is less than. Show next that honest reporting dominates any report that is greater than. Conclude that the honest report is a dominant strategy for division. The division s report determines whether or not it receives + or 0. Ideally, division would like to receive + if and only if it is nonnegative. Honest reporting insures this outcome and therefore is a dominant strategy. 1. Following the question, compare the report of to. The following table presents the payoff to division given its report and the report of division : Exercise 11 1. report 0 + 0 + 0 0 + Notice that the payoff from honest reporting is either the same or strictly more than from reporting. Honest reporting thus weakly dominates under-reporting. Now compare the report of to. The following table presents the payoff to division given its report and the report of division : report 0 0 + 0 + 0 + Notice that the payoff from honest reporting is either the same or strictly more than from reporting. Honest reporting thus weakly dominates over-reporting. 3. Does division have any other dominant strategies? Explain (your answer to 2. may be helpful). No this is shown by the above tables. 4. Assuming that each division uses its unique dominant strategy, show that the HQ provides the project exactly when it should be provided. With honest reporting, the project is provided if and only if 1 + 2, which is exactly when it should be provided. 4
5. How much of a deficit does HQ incur in following this procedure? HQ collects 2 ( 1 + 2 ) from the divisions and and funds the project at a cost of. Its deficit is therefore equal to the negative of the benefit tothefirm as a whole from the project, ( 1 + 2 ). 6. Now assume that in addition to paying when the project is provided, each division is required to pay a constant tax R regardless of whether or not the project is provided. (a) Verify that honest reporting remains the unique dominant strategy of each division. This changes division s payoff to ½ ( ) + = if + otherwise The logic of the above analysis remains the same: honest reporting insures that the division receives the larger of its two possible payoffs forall. (b) What is the deficit incurred by HQ in equilibrium? 1 + 2 ( 1 + 2 ) (c) Suppose that 1 and 2 are jointly distributed according to the distribution. 2 can be chosen so that the expected deficit of HQ is zero. This is easy: choose any 1 2 such that Show that 1 and 1 + 2 = [ 1 + 2 ] Example 12 (Pigou (1920)) A suburb is connected to a train station by two highways, one old and one modern. A large number of people in the suburb commute daily from the suburb to the station on one of these two highways. Travel time on the old highway from the suburb to the station is one hour. Travel time on the modern highway equals hours, where [0 1] is the fraction of the commuters who choose this modern highway. We make two observations. First, choosing the modern highway is a weakly dominant strategy for each commuter (it is strictly better except in the case in which every commuter chooses the modern highway). Second, there are Pareto superior outcomes. Suppose half the commuters choose the old highway and half choose the new highway. Those on the new highway have travel time equal to 1/2 hour, and so the average commuting time is 3/4 hour. It is straightfoward to show that this is the minimum possible average commuting time and x=1/2 is the unique value of x that achieves this minimum. Pigou s Example is an early example in which self-interested behavior by individuals does not lead to an efficient outcome. The Prisoner s Dilemma is another example of this phenomenon. Contrast it with the Adam Smith s idea that self-interested behavior by firms and consumers leads to an outcome that is good for everyone. You saw this idea formalized as the First Fundamental Theorem of Welfare Economics in the fall semester. Iterative Deletion of Strictly Dominated Strategies define rationality as maximization of a "reasonable" utility function MWG: The approach here is, "How far can we get in the analysis of the game from the assumption that a player acts rationally? Or instead, with the assumption that all players act rationally and know that all other players act rationally? Etc.. Dominance takes us as far as we can under the assumption that each individual player is rational. We now work with the assumption that it is common knowledge that all players are rational. This subsection begins the discussion of one player analyzing his opponents payoffs to understand what they may do in the game. Example 13 The DA s Brother vs. The Prisoner s Dilemma 12 12 0 2 10 1 2 2 10 1 1 10 5 5 1 10 5 5 5
In comparison with the Prisoner s Dilemma, here player 1 receives preferential treatment (0 2) if neither player confesses (he s the "DA s brother"). Notice that C no longer dominates DC for player 1, for he would prefer to choose DC in the event that 2 chooses DC. If player 1 knows 2 s payoffs, however, and if he knows that 2 is rational, then he can deduce that 2 would never choose DC (it is a strictly dominated strategy for him). Therefore, 1 chooses C because he deduces that 2 will also choose C. Example 14 0 3 3 2 4 1 1 2 1 0 2 1 1 1 4 3 3 2 or 0 3 3 2 4 1 1 2 1 0 2 1 1 1 4 3 3 2 Issues concerning iterative deletion: 1. Which player starts? 0 3 3 2 1 2 1 0 1 1 4 3 0 3 3 2 4 1 1 1 4 3 3 2 1 1 4 3 12 4 3 0 3 3 2 1 1 4 3 2. Is it essential that one eliminate all possible strategies at each step? 3. Can both players eliminate strategies in each iteration? 4. It doesn t solve all games (e.g., Battle of the Sexes or Meeting in New York) 1 1 4 3 12 4 3 Successive elimination of strictly dominated strategies does not depend upon the order or the thoroughness of elimination in a finite game. (Exercise 8.B.4: Consider the strategies available at stage ; forastrictly dominated strategy, there must exists a strategy that strictly dominates it that itself is not strictly dominated. The strategy will carry over to strictly dominate at a future stage.) Emphasize how the answer depends upon strict dominance. Successive elimination of weakly dominated strategies can depend on the order. Example 15 5 1 4 0 6 0 3 1 6 4 4 4 or 5 1 4 0 6 0 3 1 6 4 4 4 5 1 4 0 6 4 4 4 12 5 1 6 4 12 6 4 6 0 3 1 6 4 4 4 12 3 1 4 4 12 4 4 The rationale behind eliminating a player s weakly dominated strategy is that the opponent may use any of his strategies (i.e., one can focus on strategies by the opponents in which a given strategy is strictly dominated as grounds for eliminating it). The logic of successive elimination, however, is that the opponent may not necessarily use all of his strategies. Successively eliminating weakly dominated strategies therefore is not as sensible or coherent as a method for solving a game as successively eliminating strictly dominated strategies. Thecomplementsthefactthatitmaybe ambiguous in the solutions that it provides. Criticism of this solution concept: 1. Elimination of either weakly or strictly dominated strategies doesn t get us very far with the first price auction (we can conclude that every bidder should bid less than his value). This is a problem with these solution concepts: there are many games that they fail to solve. 2. A second criticism is that the sequence of logical deductions may exceed the abilities of most humans. 3. A third criticism is that using the solution concept requires that a player believe that his opponents are also applying it correctly. 6
Randomized or Mixed Strategies perfect recall: in an extensive form game, a player does not forget what he once knew Example 16 A game that fails to satisfy perfect recall (MWG, p. 225): notation for the set of mixed strategies: common notation for a mixed strategy: mixed vs. pure strategies note assumption of independent randomizations behavioral strategy in extensive form game: independent randomization at each information set mixed strategy: randomization over pure strategies For games of perfect recall, the two forms of randomization are equivalent Question: Is it possible to distinguish behavioral from mixed strategies in the above example of a game with imperfect recall? Let s try. We consider the following pure strategies of player 2: (R,R,L) and (L,L,R). The first letter indicates his move at the top left node, the second at the top right node, and the third at the information set. Assume player 2 plays each of these pure strategies with probability 1/2. Notice that at his information set he chooses L only if he reached this set by choosing R at the top left node and R only if he reached this set by choosing L at the top right node. His choice at the information set is clearly not independent of his choices at preceding nodes, and so we can t represent it as a behavioral strategy. Question: How do we interpret mixed strategies? 1. Does a player actually randomize? 2. Does it represent frequency within a population? Applying the notion of strict dominance to the mixed or randomized strategies of a game: ( 0 ) ( ) Notice no expected value notation. ( ) 0 denotes the probability that agent selects X ( )=1 Eliminating the strictly dominated mixed strategies in a game. 1. Eliminate the strictly dominated pure strategies (see the exercise below). 7
2. 0 strictly dominates if and only if it gives strictly more for every profile of pure strategies, ( 0 ) ( )= X Y ( ) h i ( 0 ) ( ) 6= Notice that if it holds for every profile of pure strategies of the opponents, then it also holds for every profile of mixed strategies of the opponents. Exercise 17 Exercise 8.B.6 If a pure strategy is strictly dominated, then so is any mixed strategy that plays with positive probability. Example 18 p. 241 1\2 10 1 0 4 4 2 4 3 0 5 10 2 1\2 10 1 0 4 0 5 10 2 Notice that neither player has any strictly dominated strategies, if we consider only pure strategies. The expected payoff from a 50-50 randomization of U,D, however, produces an expected payoff of 5, which strictly dominates the payoff of 4 that comes from playing M. 1\2 10 1 0 4 6 2 6 3 0 5 10 2 Note that 12 +12 is strictly dominated by even though does not dominate eitther or. Rationalizable Strategies (8.C) Bernheim and Pearce Definition 19 1. 0 is a best response to if ( 0 ) ( ) for all 2. 0 is never a best response if it is not a best response to any strategy Clearly, a player should never use a strategy that is never a best response Eliminating strategies that are never best responses eliminates all strictly dominated strategies, and perhaps more Strategies that remain after iterative elimination of strategies that are never best responses: those that a rational player can justify, or rationalize, with some reasonable conjecture concerning the behavior of his rivals (reasonable in the sense that his opponents are not presumed to play strategies that are never best responses, etc.). "Rationalizable" intuitively means that there is a plausible explanation that would justify theuseofthestrategy. 12 1 2 3 4 1 0 7 2 5 7 0 0 1 Example 20 2 5 2 3 3 5 2 0 1 3 7 0 2 5 0 7 0 1 4 0 0 0 2 0 0 10 1 Determine the set of rationalizable pure strategies. 12 1 +12 3. First eliminate 4, which is strictly dominated by 4 then strictly dominated by 2 once 4 is deleted. 12 1 2 3 12 1 0 7 2 5 7 0 1 2 3 2 5 2 3 3 5 2 1 0 7 2 5 7 0 3 7 0 2 5 0 7 2 5 2 3 3 5 2 4 0 0 0 2 0 0 3 7 0 2 5 0 7 12 1 2 3 4 1 0 7 2 5 7 0 0 1 2 5 2 3 3 5 2 0 1 3 7 0 2 5 0 7 0 1 4 0 0 0 2 0 0 10 1 8
Example 21 What bids are rationalizable in the first price auction? Suppose =max 6= so bidder can t both win the auction and make a profit. Then any bid below is a best response to the bids of the opponents. Since is arbitary in this discussion, it follows that any bid of any bidder is rationalizable. Hmmm...rationalizability doesn t help much here. Example 22 What bids are rationalizable in the second price auction? Again, any bid isabestresponsetosomeprofile of bids by the other bidders. Interestingly, the second price auction is a game in which game theorists have confidence in a unique prediction (i.e., bidding one s value by each bidder is the unique dominant strategy equilibrium). It isn t determined by rationality alone, however. Perhaps this point is helpful in understanding why subjects sometimes fail to play their unique dominant strategies in experimental tests of such procedures as the second price auction. Rationalizable strategies: A strategy is rationalizable if there is a "reasonable" conjecture concerning the behavior of opponents under which the given strategy is a best response. The "reasonable" conjecture requires that the behavior of the opponents also be best responses in this sense, and so on. = iterative elimination of strategies that are never best responses Example above: 1 2 3 are rationalizable for player 1, 1 2 3 are rationalizable for player 2 The point of rationalizability is to see how far you can go in analyzing a game based solely upon common knowledge of rationality. Let s downplay the term "common knowledge of rationality" for now, interpreting informally as "both players know that each other is rational", and so on. As the example shows, however, common knowledge of rationality may not take us very far in analyzing a game. In the remainder of Chapter 8, we ll go further by requiring some form of equilibrium behavior in the play of the game. Theorem 23 In games with =2players, the rationalizable are exactly those that survive iterative deletion of strictly dominated strategies (p. 245 of MWG). In the case of 2 players, the set of rationalizable strategies may be strictly smaller than the set of strategies that survive iterative deletion of strictly dominated strategies. For all values of, rationalizability is a different motivation for the choice and elimination of strategies from iterative deletion of strictly dominated strategies. Nash Equilibrium (sec. 8.D) Nash equilibrium: best response with a correct conjecture by each player concerning the strategies of the other players Definition 24 An -tuple ( 1 ) of pure strategies is a pure strategy Nash equilibrium if, for each player, ( ) ( 0 ) for all other pure strategies 0 of player. Nash (1951) Nash equilibrium adds to rationalizability the constraint that the players be correct in their conjectures about each others behaviors. Example 25 Meeting in NY: 12 0 0 0 0 2 pure strategy Nash equilibria 9
Mixed Strategy Nash Equilibria + : those strategies that uses with positive probability in the given strategy ( )= ( 0 ) for all 0 + ( ) ( 0 ) for all + and 0 + Example 26 Meeting in NY: 12 0 0 0 0 0 0 each player goes to GCS with probability 111 in a mixed strategy Nash equilibrium Example27(TheBraessParadox)There are 4000 motorists who drive each morning from the point labeled Start to the point labeled Finish. There are two possible routes, one through and one through. The routes and can be thought of as bridges or limited capacity roads. The travel time for each motorist on each of these routes is minutes, where is the total number of motorists who choose that particular route. Travel time thus increases linearly in the number of motorists who choose a particular route. The routes and are high capacity, modern roads that are each sufficiently large to handle all 4000 motorists without increasing the travel time. The travel time on these routes is 45 minutes, regardless of how many motorists travel on the route. We assume that each motorist wishes to minimize his total travel time from Start to Finish, taking into account the travel pattern determined by the routes chosen by all other motorists. We thus interpret this as a game with 4000 players in which player chooses either the top route through or the bottom route through. Each motorist therefore has two possible strategies. A motorist will change his route in favor of a shorter trip. We thus look for a distribution of motorists across the two routes that forms a Nash equilibrium, i.e., no motorist can benefit by changing routes, given the choices of every other motorist. We first characterize a property of a Nash equilibrium. We seek a number of motorists for the top route and a number of motorists for the bottom route, where + = 4000 The travel time on the top route is +45 and the travel time on the bottom route is +45 A motorist who switches from the top route to the bottom route changes his travel time from +45 10
to +1 +45 because he adds a motorist on the bottom route. For the driver on the top route to have no incentive to switch, it must be the case that +1 +45 +45 +1 Similarly, for a driver on the bottom to have no incentive to switch, it must be the case that +1 +45 +45 +1 For a Nash equilibrium, it is necessary that no driver want to switch, i.e., both of these inequalities hold. Therefore, +1 1 The number equals either 1,,or +1. Recall that there are 4000 motorists, and so + = 4000 If = 1, then + =2 1 = 4000 which contradicts being a whole number. Similarly, = +1 is not possible, and so = = 2000 is the only possibility for a Nash equilibrium. It is clear that this indeed is a Nash equilibrium distribution of motorists, for a driver who changes routes strictly increases his travel time. Each motorist s travel time in the only Nash equilibrium is 2000 +45=65 minutes. There s no paradox yet, but here it comes! Suppose next that in the interest of improving traffic flow a one-way route is added from to : For simplicity, we ll assume that travel time on the route equals zero. How does the addition of this "shortcut" change the travel time of motorists in a Nash equilibrium? We claim that the route is the unique dominant strategy of every motorist in this new game of choosing one s route. To verify this, we select a motorist and consider his choice of a route given that the choices of the other 3999 motorists determine a value of along the route and a value along the route 11
. It is not necessarily the case that + = 3999, because some of the other motorists may take the shortcut and thus count among both the numbers and. It is true, however, that, 3999 and + 3999. The selected motorist now has 3 possible routes with 3 possible travel times: route travel time +1 +45 +1 + +1 +1 +45 The "+1" indicates the congestion that the selected motorist creates by adding himself to a particular route. We have +1 + +1 +1 + 4000 = +1 +40 +1 +45 and similarly, +1 + +1 4000 + +1 =40+ +1 +1 +45 The route is therefore fastest for the selected motorist regardless of the decisions of the other motorists and the values of and that are determined by these decisions. It is therefore a dominant strategy for each motorist. The unique dominant strategy equilibrium outcome is therefore that all motorists select the route. The driving time of each motorist is then 4000 + 4000 =80 minutes, which is strictly more than the 65 minutes required before the shortcut was introduced. This is the Braess paradox, namely, adding a shortcut can increase the average travel time. Conversely, the average travel time may be decreased by closing a road! More generally, we can apply this to congestion in any kind of network in which users choose their own routes. Adding a link in a network can diminish the performance of the network while deleting a link can improve performance. This depends upon the assumption that network users array themselves as in a Nash equilibrium. Questions about MSNE: 1. Why do players bother to randomize when it doesn t alter their expected payoffs? 2. Equilibrium depends upon randomization according to precise probabilities. Do we believe that people behave in this way? 3. Independence of randomization vs. correlated equilibrium (Aumann). Discussion of Nash Equilibrium: Why should we expect the players to play a Nash equilibrium? This remains an active and incomplete area of research. 1. Nash equilibrium as a consequence of rational inference. But as we saw, rationalizability is the consequence of common knowledge of rationality and the structure of the game, and it does not necessarily lead to correct conjectures on the part of the players 2. Nash equilibrium as a necessary condition if there is a unique predicted outcome of the game. Q: Why would players believe that there is a unique way to play the game? It isn t a consequence of rationality. 3. Focal points e.g., meeting in NY. As with 2., a focal point would have to be a Nash equilibrium. 4. Nash equilibrium as a self-enforcing agreement. Nonbinding communication before the game. But shouldn t the process of communication be modeled, and don t the players communicate strategically? V. Smith: If players can communicate, then they will act cooperatively (not Nash). 12
5. Nash equilibrium as a stable social convention. E.g., which side of the sidewalk to walk on. Stability of the social convention requires Nash equilibrium. Binmore s parable of the quadratic equation 13