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Solution Manual for

A Course in Game Theory

This manual was typ eset by the authors, who are greatly indebted to Donald Knuth (the creator of TEX), Leslie Lamp ort (the creator of LATEX), and Eb erhard Mattes (the creator of emTEX) for generously putting sup erlative software in the public domain, and to Ed Sznyter for providing critical help with the macros we use to execute our numb ering scheme.

Version 1.1, 97/4/

Contents

• 2 Nash Equilibrium Preface xi
  • Exercise 18.2 (First price auction )
  • Exercise 18.3 (Second price auction )
  • Exercise 18.5 (War of attrition )
  • Exercise 19.1 (Location game )
  • Exercise 20.2 (Necessity of conditions in Kakutani's theorem )
  • Exercise 20.4 (Symmetric games )
  • Exercise 24.1 (Increasing payo s in strictly competitive game )
  • Exercise 27.2 (BoS with imperfect information )
  • Exercise 28.1 (Exchange game )
  • Exercise 28.2 (More information may hurt )
• 3 Mixed, Correlated, and Evolutionary Equilibrium
  • Exercise 35.1 (Guess the average )
  • Exercise 35.2 (Investment race )
  • Exercise 36.1 (Guessing right )
  • Exercise 36.2 (Air strike )
  • Exercise 36.3 (Technical result on convex sets )
  • Exercise 42.1 (Examples of Harsanyi's puri cation )
  • Exercise 48.1 (Example of correlated equilibrium )
  • Exercise 51.1 (Existence of ESS in 2  2 game )
  • Actions 4 Rationalizability and Iterated Elimination of Dominated
  • Exercise 56.3 (Example of rationalizable actions )
  • Exercise 56.4 (Cournot duopoly )
  • Exercise 56.5 (Guess the average ) vi Contents
  • Exercise 57.1 (Modi ed rationalizability in location game )
  • Exercise 63.1 (Iterated elimination in location game )
  • Exercise 63.2 (Dominance solvability )
  • Exercise 64.1 (Announcing numbers )
  • Exercise 64.2 (Non-weakly dominated action as best response )
• 5 Knowledge and Equilibrium
  • Exercise 69.1 (Example of information function )
  • Exercise 69.2 (Remembering numbers )
  • Exercise 71.1 (Information functions and know ledge functions )
  • Exercise 71.2 (Decisions and information )
  • Exercise 76.1 (Common know ledge and di erent beliefs )
  • Exercise 76.2 (Common know ledge and beliefs about lotteries )
  • Exercise 81.1 (Know ledge and correlated equilibrium )
• 6 Extensive Games with Perfect Information
  • Exercise 94.2 (Extensive games with 2  2 strategic forms )
  • Exercise 98.1 (SPE of Stackelberg game )
  • Exercise 99.1 (Necessity of nite horizon for one deviation property )
  • Exercise 100.1 (Necessity of niteness for Kuhn's theorem )
  • Exercise 100.2 (SPE of games satisfying no indi erence condition )
  • Exercise 101.1 (SPE and unreached subgames )
  • Exercise 101.2 (SPE and unchosen actions )
  • Exercise 101.3 (Armies )
  • Exercise 102.1 (ODP and Kuhn's theorem with chance moves )
  • Exercise 103.1 (Three players sharing pie )
  • Exercise 103.2 (Naming numbers )
  • Exercise 103.3 (ODP and Kuhn's theorem with simultaneous moves )
  • Exercise 108.1 (-equilibrium of centipede game )
  • Exercise 114.1 (Variant of the game Burning money )
  • Exercise 114.2 (Variant of the game Burning money )
• 7 A Mo del of Bargaining
  • Exercise 123.1 (One deviation property for bargaining game )
  • Exercise 125.2 (Constant cost of bargaining )
  • Exercise 127.1 (One-sided o ers )
  • Exercise 128.1 (Finite grid of possible o ers )
  • Exercise 129.1 (Outside options )
  • Exercise 130.2 (Risk of breakdown ) Contents vii
  • Exercise 131.1 (Three-player bargaining )
• 8 Rep eated Games
  • Exercise 139.1 (Discount factors that di er )
  • Exercise 143.1 (Strategies and nite machines )
  • Exercise 144.2 (Machine that guarantees vi )
  • Exercise 145.1 (Machine for Nash folk theorem )
  • Exercise 146.1 (Example with discounting )
  • Exercise 148.1 (Long- and short-lived players )
  • Exercise 152.1 (Game that is not ful l dimensional )
  • Exercise 153.2 (One deviation property for discounted repeated game )
  • Exercise 157.1 (Nash folk theorem for nitely repeated games )
• 9 Complexity Considerations in Rep eated Games
  • Exercise 169.1 (Unequal numbers of states in machines )
  • Exercise 173.1 (Equilibria of the Prisoner's Dilemma)
  • Exercise 173.2 (Equilibria with introductory phases )
  • Exercise 174.1 (Case in which constituent game is extensive game )
• 10 Implementation Theory
  • Exercise 182.1 (DSE-implementation with strict preferences )
  • Exercise 183.1 (Example of non-DSE implementable rule )
  • Exercise 185.1 (Groves mechanisms )
  • Exercise 191.1 (Implementation with two individuals )
• 11 Extensive Games with Imp erfect Information
  • Exercise 203.2 (De nition of Xi (h))
  • Exercise 208.1 (One-player games and principles of equivalence )
  • Exercise 216.1 (Example of mixed and behavioral strategies )
  • Exercise 217.1 (Mixed and behavioral strategies and imperfect recal l )
  • Exercise 217.2 (Splitting information sets )
  • Exercise 217.3 (Parlor game )
• 12 Sequential Equilibrium
  • Exercise 226.1 (Example of sequential equilibria )
  • Exercise 227.1 (One deviation property for sequential equilibrium )
  • Exercise 229.1 (Non-ordered information sets )
  • Exercise 234.2 (Sequential equilibrium and PBE )
  • Exercise 237.1 (Bargaining under imperfect information ) viii Contents
  • Exercise 238.1 (PBE is SE in Spence's model )
  • Exercise 243.1 (PBE of chain-store game )
  • Exercise 246.2 (Pre-trial negotiation )
  • Exercise 252.2 (Trembling hand perfection and coalescing of moves )
  • Exercise 253.1 (Example of trembling hand perfection )
• 13 The Core
  • Exercise 259.3 (Core of production economy )
  • Exercise 260.2 (Market for indivisible good )
  • Exercise 260.4 (Convex games )
  • Exercise 261.1 (Simple games )
  • Exercise 261.2 (Zerosum games )
  • Exercise 261.3 (Pol lute the lake )
  • Exercise 263.2 (Game with empty core )
  • Exercise 265.2 (Syndication in a market )
  • Exercise 267.2 (Existence of competitive equilibrium in market )
  • Exercise 268.1 (Core convergence in production economy )
  • Exercise 274.1 (Core and equilibria of exchange economy )
• 14 Stable Sets, the Bargaining Set, and the Shapley Value
  • Exercise 280.1 (Stable sets of simple games )
  • Exercise 280.2 (Stable set of market for indivisible good )
  • Exercise 280.3 (Stable sets of three-player games )
  • Exercise 280.4 (Dummy's payo in stable sets )
  • Exercise 280.5 (Generalized stable sets )
  • Exercise 283.1 (Core and bargaining set of market )
  • Exercise 289.1 (Nucleolus of production economy )
  • Exercise 289.2 (Nucleolus of weighted majority games )
  • Exercise 294.2 (Necessity of axioms for Shapley value )
  • Exercise 295.1 (Example of core and Shapley value )
  • Exercise 295.2 (Shapley value of production economy )
  • Exercise 295.4 (Shapley value of a model of a parliament )
  • Exercise 295.5 (Shapley value of convex game )
  • Exercise 296.1 (Coalitional bargaining )
• 15 The Nash Bargaining Solution
  • Exercise 309.1 (Standard Nash axiomatization )
  • Exercise 309.2 (Eciency vs. individual rationality )

Preface

This manual contains solutions to the exercises in A Course in Game Theory by Martin J. Osb orne and Ariel Rubinstein. (The sources of the problems are given in the section entitled \Notes" at the end of each chapter of the b o ok.) We are very grateful to Wulong Gu for correcting our solutions and providing many of his own and to Ebb e Hendon for correcting our solution to Exercise 227.1. Please alert us to any errors that you detect.

Errors in the Book Postscript and PCL les of errors in the b o ok are kept at http://www.socsci.mcmaster.ca/~econ/faculty/osborne/cgt/

Martin J. Osborne [email protected] Department of Economics, McMaster University Hamilton, Canada, L8S 4M

Ariel Rubinstein [email protected] Department of Economics, Tel Aviv University Ramat Aviv, Israel, 69978 Department of Economics, Princeton University Princeton, NJ 08540, USA

2 Nash^ Equilibr^ iu^ m

18.2 (First price auction ) The set of actions of each player i is [0; 1 ) (the set of p ossible bids) and the payo of player i is vi bi if his bid bi is equal to the highest bid and no player with a lower index submits this bid, and 0 otherwise. The set of Nash equilibria is the set of pro les b of bids with b 1 2 [v 2 ; v 1 ], bj  b 1 for all j 6 = 1, and bj = b 1 for some j 6 = 1. It is easy to verify that all these pro les are Nash equilibria. To see that there are no other equilibria, rst we argue that there is no equilibrium in which player 1 do es not obtain the ob ject. Supp ose that player i 6 = 1 submits the highest bid bi and b 1 < bi. If bi > v 2 then player i's payo is negative, so that he can increase his payo by bidding 0. If bi  v 2 then player 1 can deviate to the bid bi and win, increasing his payo. Now let the winning bid b e b^. We have b^  v 2 , otherwise player 2 can change his bid to some value in (v 2 ; b^ ) and increase his payo. Also b^  v 1 , otherwise player 1 can reduce her bid and increase her payo. Finally, bj = b for some j 6 = 1 otherwise player 1 can increase her payo by decreasing her bid.

Comment An assumption in the exercise is that in the event of a tie for the highest bid the winner is the player with the lowest index. If in this event the ob ject is instead allo cated to each of the highest bidders with equal probability then the game has no Nash equilibrium. If ties are broken randomly in this fashion and, in addition, we deviate from the assumptions of the exercise by assuming that there is a nite numb er of p ossible bids then if the p ossible bids are close enough together there is a Nash equilibrium in which player 1's bid is b 1 2 [v 2 ; v 1 ] and one of the other players' bids is the largest p ossible bid that is less than b 1. Note also that, in contrast to the situation in the next exercise, no player has a dominant action in the game here.

2 Chapter 2. Nash Equilibrium

18.3 (Second price auction ) The set of actions of each player i is [0; 1 ) (the set of p ossible bids) and the payo of player i is vi bj if his bid bi is equal to the highest bid and bj is the highest of the other players' bids (p ossibly equal to bi ) and no player with a lower index submits this bid, and 0 otherwise. For any player i the bid bi = vi is a dominant action. To see this, let xi b e another action of player i. If maxj 6 =i bj  vi then by bidding xi player i either do es not obtain the ob ject or receives a nonp ositive payo , while by bidding bi he guarantees himself a payo of 0. If maxj 6 =i bj < vi then by bidding vi player i obtains the go o d at the price maxj 6 =i bj , while by bidding xi either he wins and pays the same price or loses. An equilibrium in which player j obtains the go o d is that in which b 1 < vj , bj > v 1 , and bi = 0 for all players i 2 = f 1 ; j g.

18.5 (War of attrition ) The set of actions of each player i is Ai = [0; 1 ) and his payo function is

ui (t 1 ; t 2 ) =

ti if ti < tj vi = 2 ti if ti = tj vi tj if ti > tj where j 2 f 1 ; 2 g n fig. Let (t 1 ; t 2 ) b e a pair of actions. If t 1 = t 2 then by conceding slightly later than t 1 player 1 can obtain the ob ject in its entirety instead of getting just half of it, so this is not an equilibrium. If 0 < t 1 < t 2 then player 1 can increase her payo to zero by deviating to t 1 = 0. Finally, if 0 = t 1 < t 2 then player 1 can increase her payo by deviating to a time slightly after t 2 unless v 1 t 2  0. Similarly for 0 = t 2 < t 1 to constitute an equilibrium we need v 2 t 1  0. Hence (t 1 ; t 2 ) is a Nash equilibrium if and only if either 0 = t 1 < t 2 and t 2  v 1 or 0 = t 2 < t 1 and t 1  v 2. Comment An interesting feature of this result is that the equilibrium out- come is indep endent of the players' valuations of the ob ject.

19.1 (Location game ) 1 There are n players, each of whose set of actions is fOut g [ [0; 1]. (Note that the mo del di ers from Hotelling's in that players cho ose whether or not to b ecome candidates.) Each player prefers an action pro le in which he obtains more votes than any other player to one in which he ties for the largest numb er of votes; he prefers an outcome in which he ties for (^1) Correction to rst printing of book : The rst sentence on page 19 of the b o ok should b e amended to read \There is a continuum of citizens, each of whom has a favorite p osition; the distribution of favorite p ositions is given by a density function f on [0; 1] with f (x) > 0 for all x 2 [0; 1]."

4 Chapter 2. Nash Equilibrium

20.2 (Necessity of conditions in Kakutani's theorem )

i. X is the real line and f (x) = x + 1. ii. X is the unit circle, and f is rotation by 90 ^. iii. X = [0; 1] and

f (x) =

f 1 g if x < (^12) f 0 ; 1 g if x = (^12) f 0 g if x > 12.

iv. X = [0; 1]; f (x) = 1 if x < 1 and f (1) = 0.

20.4 (Symmetric games ) De ne the function F : A 1! A 1 by F (a 1 ) = B 2 (a 1 ) (the b est resp onse of player 2 to a 1 ). The function F satis es the conditions of Lemma 20.1, and hence has a xed p oint, say a 1. The pair of actions (a 1 ; a 1 ) is a Nash equilibrium of the game since, given the symmetry, if a 1 is a b est resp onse of player 2 to a 1 then it is also a b est resp onse of player 1 to a 1. A symmetric nite game that has no symmetric equilibrium is Hawk{Dove (Figure 17.2). Comment In the next chapter of the b o ok we intro duce the notion of a mixed strategy. From the rst part of the exercise it follows that a nite symmetric game has a symmetric mixed strategy equilibrium.

24.1 (Increasing payo s in strictly competitive game )

a. Let ui b e player i's payo function in the game G, let wi b e his pay- o function in G^0 , and let (x^ ; y ^ ) b e a Nash equilibrium of G^0. Then, us- ing part (b) of Prop osition 22.2, we have w 1 (x^ ; y ^ ) = miny maxx w 1 (x; y )  miny maxx u 1 (x; y ), which is the value of G. b. This follows from part (b) of Prop osition 22.2 and the fact that for any function f we have maxx 2 X f (x)  maxx 2 Y f (x) if Y  X. c. In the unique equilibrium of the game

Chapter 2. Nash Equilibrium 5

player 1 receives a payo of 3, while in the unique equilibrium of

she receives a payo of 2. If she is prohibited from using her second action in this second game then she obtains an equilibrium payo of 3, however.

27.2 (BoS with imperfect information ) The Bayesian game is as follows. There are two players, say N = f 1 ; 2 g, and four states, say = f(B ; B ); (B ; S ); (S; B ); (S; S )g, where the state (X ; Y ) is interpreted as a situation in which player 1's preferred comp oser is X and player 2's is Y. The set Ai of actions of each player i is fB ; S g, the set of signals that player i may receive is fB ; S g, and player i's signal function i is de ned by i (! ) = !i. A b elief of each player i is a probability distribution pi over. Player 1's preferences are those represented by the payo function de ned as follows. If! 1 = B then u 1 ((B ; B );! ) = 2, u 1 ((S; S );! ) = 1, and u 1 ((B ; S );! ) = u 1 ((S; B );! ) = 0; if! 1 = S then u 1 is de ned analogously. Player 2's preferences are de ned similarly. For any b eliefs the game has Nash equilibria ((B ; B ); (B ; B )) (i.e. each typ e of each player cho oses B ) and ((S; S ); (S; S )). If one player's equilibrium action is indep endent of his typ e then the other player's is also. Thus in any other equilibrium the two typ es of each player cho ose di erent actions. Whether such a pro le is an equilibrium dep ends on the b eliefs. Let qX = p 2 (X ; X )=[p 2 (B ; X ) + p 2 (S; X )] (the probability that player 2 assigns to the event that player 1 prefers X conditional on player 2 preferring X ) and let pX = p 1 (X ; X )=[p 1 (X ; B ) + p 1 (X ; S )] (the probability that player 1 assigns to the event that player 2 prefers X conditional on player 1 preferring X ). If, for example, pX  13 and qX  13 for X = B , S , then ((B ; S ); (B ; S )) is an equilibrium.

28.1 (Exchange game ) In the Bayesian game there are two players, say N = f 1 ; 2 g, the set of states is = S  S , the set of actions of each player is fExchange ; Don't exchange g, the signal function of each player i is de ned by i (s 1 ; s 2 ) = si , and each player's b elief on is that generated by two inde- p endent copies of F. Each player's preferences are represented by the payo

3 Mixed,^ Correlated,^ and^ Evolutionary

Equilib ri um

35.1 (Guess the average ) Let k ^ b e the largest numb er to which any player's strat- egy assigns p ositive probability in a mixed strategy equilibrium and assume that player i's strategy do es so. We now argue as follows.

 In order for player i's strategy to b e optimal his payo from the pure strategy k ^ must b e equal to his equilibrium payo.

 In any equilibrium player i's exp ected payo is p ositive, since for any strategies of the other players he has a pure strategy that for some re- alization of the other players' strategies is at least as close to 23 of the average numb er as any other player's numb er.

 In any realization of the strategies in which player i cho oses k ^ , some other player also cho oses k ^ , since by the previous two p oints player i's payo is p ositive in this case, so that no other player's numb er is closer to 23 of the average numb er than k ^. (Note that all the other numb ers cannot b e less than 23 of the average numb er.)

 In any realization of the strategies in which player i cho oses k ^  1, he can increase his payo by cho osing k ^ 1, since by making this change he b ecomes the outright winner rather than tying with at least one other player.

The remaining p ossibility is that k ^ = 1: every player uses the pure strategy in which he announces the numb er 1.

8 Chapter 3. Mixed, Correlated, and Evolutionary Equilibrium

35.2 (Investment race ) The set of actions of each player i is Ai = [0; 1]. The payo function of player i is

ui (a 1 ; a 2 ) =

ai if ai < aj 1 2 ^ ai^ if^ ai^ =^ aj 1 ai if ai > aj ,

where j 2 f 1 ; 2 g n fig. We can represent a mixed strategy of a player i in this game by a probability distribution function Fi on the interval [0; 1], with the interpretation that Fi (v ) is the probability that player i cho oses an action in the interval [0; v ]. De ne the support of Fi to b e the set of p oints v for which Fi (v + ) Fi (v ) > 0 for all  > 0, and de ne v to b e an atom of Fi if Fi (v ) > lim# 0 Fi (v ). Supp ose that (F 1  ; F 2  ) is a mixed strategy Nash equilibrium of the game and let S (^) i b e the supp ort of F (^) i for i = 1, 2. Step. S 1  = S 2 . Proof. If not then there is an op en interval, say (v ; w ), to which F (^) i assigns p ositive probability while F (^) j assigns zero probability (for some i, j ). But then i can increase his payo by transferring probability to smaller values within the interval (since this do es not a ect the probability that he wins or loses, but increases his payo in b oth cases). Step. If v is an atom of F (^) i then it is not an atom of F (^) j and for some  > 0 the set S (^) j contains no p oint in (v ; v ). Proof. If v is an atom of F (^) i then for some  > 0, no action in (v ; v ] is optimal for player j since by moving any probability mass in F (^) i that is in this interval to either v +  for some small  > 0 (if v < 1) or 0 (if v = 1), player j increases his payo. Step. If v > 0 then v is not an atom of F (^) i for i = 1, 2. Proof. If v > 0 is an atom of F (^) i then, using Step 2, player i can increase his payo by transferring the probability attached to the atom to a smaller p oint in the interval (v ; v ). Step. S (^) i = [0; M ] for some M > 0 for i = 1, 2. Proof. Supp ose that v 2 = S (^) i and let w ^ = inf fw : w 2 S (^) i and w  v g > v. By Step 1 we have w ^2 S (^) j , and hence, given that w ^ is not an atom of F (^) i by Step 3, we require j 's payo at w ^ to b e no less than his payo at v. Hence w ^ = v. By Step 2 at most one distribution has an atom at 0, so M > 0.