# Game Theory EconC31 - University of Warwick EC941 - Game Theory Lecture 1 Francesco Squintani Email: [email protected] 1 Syllabus 1. Games in Strategic Form Definition and Solution Concepts Applications Readings: Chapter 2, 3, 12 2. Mixed Strategies Nash Equilibrium and Rationalizability Correlated Equilibrium Readings: Chapter 4 2

3. Bayesian Games Definition Information and Bayesian Games Cournot Duopoly and Public Good Provision Readings: Sections 9.1 to 9.6 4.Bayesian Game Applications Juries and Information Aggregation Auctions with Private Information Readings: Sections 9.7 to 9.8 3 5. Extensive-Form Games Definition Subgame Perfection and Backward Induction Applications Readings: Chapters 5, 6 and 7

6. Extensive-Form Games with Imperfect Information Definition Spence Signalling Game Crawford and Sobel Cheap Talk Readings: Chapter 10 4 7. Repeated Games Infinitely Repeated Games Nash and Subgame-Perfect Equilibrium Finitely Repeated Games Readings: Chapter 14 and 15 8. Bargaining Ultimatum Game and Hold Up Problem Rubinstein Alternating Offer Bargaining Nash Axiomatic Bargaining Readings: Section 6.2 and Chapter 16

5 9. Review Session Solution of Past Exam Questions Reference: An Introduction to Game Theory Martin J. Osborne, Oxford University Press, 2003. Assessment: Final Exam (100% of the grade) Office Hours: Wednesday 9:00-11:00 6 Structure of the Lecture

Definition of Games in Strategic Form. Solution Concepts Nash Equilibrium, Dominance and Rationalizability. Applications Cournot Oligopoly, Bertrand Duopoly, Downsian Electoral Competition, Vickrey Second Price Auction. 7 What is Game Theory?

Game Theory is the formal study of strategic interactions. A strategic interaction involves two or more agents. They maximize their payoffs and are aware that their opponents maximize payoffs. Applications range from economics to politics, to biology and computer science. 8 Games in Strategic Form A game in strategic form is

a set of players: {1, 2, , I} for each player i, a set Si of strategies si for each player i, preferences over the set of strategy profiles S={(s1 , , sI )}, represented by u : S RI (a strategy profile includes one strategy for each player). 9 Solution Concepts A solution concept is a mathematical rule to find the solution of a game.

It allows the modeler to formulate a prediction on the play of the interaction she modeled as a game. Today, we will study 3 solution concepts: Nash Equilibrium Dominance Rationalizability 10 Nash Equilibrium A (pure-strategy) Nash equilibrium is a strategy profile s with the property that no player i can do

better by choosing a strategy different from si, given that every other player j adheres to sj. Definition The strategy profile s is a Nash equilibrium if, for every player i, ui(s) ui(si, si) for 11 There are two main justifications for Nash Equilibrium: Self-Enforcing Contract. The players meet and agree before playing on the course of actions s. The contract s is self-enforcing if no player has reasons to deviate if the others do not.

Learning Equilibrium Play. The play s is in equilibrium if no player i would deviate, were she to learn the opponents play s-i, because of 12 Dominance A players strategy strictly dominates another one if it gives a higher payoff, no matter of what other players do. Definition Player i s strategy si strictly dominates strategy si if ui (si , si) > ui (si , si) for every profile si of opponents strategies. 13

A players strategy weakly dominates another strategy if it is always at least as good, and sometimes better. Definition Player i s strategy si weakly dominates her strategy si if ui (si , si) ui(si , si) for every profile si of opponents strategies ui(si , si) > ui(si , si) for some profile si of opponents strategies. Note There exist games with Nash Equilibria s that include weakly dominated 14 strategies si for some player i. Player 2 C Player 1

A B D 1, 1 0, 0 0, 0 0, 0 There are two Nash Equilibria: (A,C) and (B,D). The Nash Equilibrium (B,D) is weakly 15

Rationalizability Rationalizability is defined via iterated deletion of strictly dominated strategies. Consider a finite game G = (I, S, u). For each player i, and round t = 1, . . . , T, iteratively define the set Xit of strategies of player i as follows. 16 For each t = 0, . . . , T 1, Xit+1 is a subset of Xit such that every strategy of player i in Xit that is not in Xit+1 is strictly dominated in the game where the set of strategy of each player j is reduced to Xjt (in each round, delete all strictly dominated

strategies). The final index T is such that no strategy in XiT is strictly dominated in the game where the set of strategies of each player j is reduced to XjT (proceed until no strategy is strictly 17 Rationalizability is justified by common knowledge of rationality. Each player is rational: She does not play strictly dominated strategies. Each player knows that every player is rational: She can reduce the game by deleting all players strictly dominated

strategies from her model of the interaction (the game). Each player knows that every player knows that every player is rational: She deletes all strictly dominated strategies in the reduced 18 Best Response Correspondences The best response correspondence Bi of player i assigns to each profile s-i of opponents strategies, the set of player i s strategies that maximizes her payoff. Definition The best response correspondence Bi of player i is: Bi (si) = {si in Si : ui(si, si) ui(si, si) for all si in Si}. Proposition The strategy profile s is a

19 Prisoners Dilemma Two prisoners are separately interviewed. By accusing the other suspect, ones prison term is reduced. But if they both stayed quiet, they would not be incarcerated. Players: The two suspects. Strategies: Each players set of strategy is {Quiet, Fink}.

Preferences: Suspect 1s ordering of the 20 Suspect 2 Quiet Fink Suspect 1 Quiet Fink 2, 2 0, 3

3, 0 1, 1 21 Solutions of Prisoners Dilemma Quiet Quiet Fink Fink 2, 2 0, 3

3, 0 1, 1 k is the best response of each player, regardless of w other player does. Fink is the strictly dominant and onalizable strategy. (Fink, Fink) is the Nash Equilib 22 Bach or Stravinsky Two daters would rather be together than separate, but dater 1 prefers Bach and dater 2 prefers Stravinsky. Players: The two daters.

Strategies: Each daters strategy set is {Bach, Stravinsky}. Preferences: Dater 1s ordering of the strategy profiles, from best to worst is (B,23 Dater 2 Dater 1 Bach Bach Stravinsky Stravinsky

2, 1 0, 0 0, 0 1, 2 hey can coordinate, either the two daters go to Bach cert or to Stravinskys concert. 24 Solutions of Bach or Bach Stravinsky Stravinsky Bach Stravinsky

2, 1 0, 0 0, 0 1, 2 For each player, B is the best response to B, and S is the best response to S. There are two Nash Equilibria, (B, B) and (S, S). All strategies are rationalizable, and none is dominated. 25 Matching Pennies Player 1 wins if the coins are matched. Player 2 wins if they are not matched.

Players: The two players. Strategies: Each players set of actions is {Head, Tail}. Preferences: Player 1s ordering of the strategy profiles, from best to worst, is (H, T) = (T, H), (H, H) = (T, T). Player 2s ordering is (H, H) = (T, T), (H, 26 Player 2 Player 1 Head Tail

Head Tail -1, 1 1, -1 1, -1 -1, 1 ere is no sure way to win for either of the players. 27 Solutions of Matching Pennies Head

Head Tail Tail -1, 1 1, -1 1, -1 -1, 1 For player 1, H is the best response to T and viceversa, for player 2, H is the best response to H and T is the best response to T. All strategies are rationalizable 28

Cournot Oligopoly A good is produced by n firms. Firm is cost of producing qi units is Ci(qi). Ci is an increasing function. The firms' total output is Q = q1 + + qn. The market price is P(Q). P is the inverse demand function,

decreasing if positive. 29 Linear Costs and Demand Firm is revenue is qi P(q1 + + qn). Firm is profit is revenue minus cost: pi(q1 + + qn) = qi P(q1 + + qn) - Ci (qi). Ci (qi) = cqi, i=1, , n.

P (Q) = a - Q if a > Q, P(Q) = 0 if a < Q. pi(q1, , qn) = qi [a (q1 + + qn)] - cqi. 30 To find the Best Response Functions, differentiate pi with respect to qi, set it equal to zero, and obtain: dpi (q1, , qn)/dqi = a qi (q1 + + qn) c = 0. Best Response functions:

bi (qi) = [a (q1 + + qi-1 + qi+1 ++ q n) c]/2. To find the Nash equilibria, we solve the system of best-response functions. Because this system is linear and symmetric, we equalize qi* across i = 1, 31 Solving the above equation, we find that the Nash equilibrium quantity is:

qi* = [a c]/(n+1). Substituting in the formula for the price, we find that the Nash equilibrium price is: pi (qi*) = a Q* = a n[a c]/(n+1) = [a + nc]/(n+1). The Nash equilibrium profits are: pi (qi*) = qi*[a Q*] cqi* = [a c ]2/(n-1)2. 32 q2 With n = 2,

b1(q2) = [a q2 c]/2. b2(q1) = [a q1 c]/2. b1(q2) [a c]/3 (q1*, q2*) qi* = [a c]/3, i = 1, b2(q1) [a c]/3 q1 33 Bertrand Competition

Unlike Cournot competition, firms compete in prices. The demand function is denoted by D, if the good is available at the price p, then the total amount demanded is D(p). The firm setting the lowest price sells to all the market. 34 Linear Costs and Demand

Ci(qi) = cqi, i=1, , n. D(p) = a p if a > p, D(p) = 0 if a < p. Let pi = min {pj, j different from i}. The profit is: pi(p1, , pn) = (pi c)(a - pi) if pi < pi, pi(p) = (pi c)(a - pi)/|{k : pk = pi}| if pi = pi, pi(p) = 0

if pi > pi. 35 Best-Response Correspondence Suppose that there only two firms, so that pi= pj. If pj < c, then pi(p) < 0 for pi < pj, pi pi(p) = 0 for pi > pj : bi(p) = {pi : pi > pj}. pj c pm

pi 36 If pj = c, then pi(p) < 0 for pi < pj, pi(p) = 0 for pi > pj : bi(p) = {pi : pi > pj}. If pj > pm, then bi(p) = {pm}. pi pi pj c

pm pi c p m pj pi 37 If c < pj < pm then pi(p) increases in pj, but discontinuously drops at pi = pj. So, bi(p) = f. The best response correspondence is pi empty.

c pj pi 38 In sum, the best-response correspondence is: b (p) = {p : p > p }, if p < c, i i i j j

bi(p) = {pi : pi > pj}, if pj = c, bi(p) = f, if c < pj < pm, bi(p) = {pm}, if pj > pm. The Nash equilibrium is pi = c, for all i = 1, ,n. Intuitively, selling at any price pi < c yields negative profit. 39 If the lowest industry price were p > c, then Downsian Electoral Competition

The players are 2 candidates in an election. A strategy is a real number x, representing a policy on the left-right political ideology spectrum. After the candidates choose policies, each citizen votes for the candidate with the policy she prefers. 40

The voters are a continuum with diverse ideologies, with cumulative distribution F. For any k, a voter with ideology y is indifferent between the policies y - k and y + k. The median m is such that 1/2 of voters has ideologies y > m, and 1/2 has ideologies y < m. So, F (m) = 1/2. 41 Best Response Functions

Fix the policy x2 of candidate 2 and consider 1s choice. Suppose that x2 < m, the case for x2 > m is symmetric. If candidate 1 chooses x1 < x2 then she wins the votes of citizens with ideology y < ( x1 + x2 ). Because ( x1 + x2 ) < x2 < m, it follows

that 42 If x1 > x2, then candidate 1 wins the votes of citizens with ideology y > ( x1 + x2 ). She wins more or less than of the votes if and only if 1 F( ( x1 + x2 )) > . In this case, she wins the election. This is equivalent to ( x1 + x2 ) < m, i.e. x1 < 2m - x2.

So, b1 (x2) = {x1 : x2 < x1 < 2m - x2 } for x2 < m. 43 For x2 > m, b1 (x2) = {x1 : 2m - x2 < x1 < x2}. If x2 = m, then player 1 loses the election unless she plays x1 = m. So b1 (m) = {m}. By using the best response

correspondences the unique Nash Equilibrium is (m, m). The candidates political platforms converge to the median policy. 44 Intuitively, consider any pair of platforms (x1, x2) other than (m, m). One candidate can win the election by deviating and locating e closer to m than x2. Hence (x1, x2) is not a Nash Equilibrium. If instead x1=m, then candidate 2 loses the election for sure unless she plays x2 = m.

Hence (m, m) is a Nash Equilibrium. 45 Vickrey Second-Price Auctions In an English auction, n bidders submit increasing bids for a good, until only one is left, who wins the auction. The price paid by the last bidder is her last bid.

Suppose each bidders valuation of the good is independent of the other bidders values. For example, Vickreys model applies when the good is a work of art, but not 46 The English auction is equivalent to a sealed-bid auction, in which each bidder decides, before bidding begins, the most she is willing to bid. To win, the bidder with the highest valuation needs to bid slightly more than the second highest maximal bid.

If the bidding increment is small, the price the winner pays equals the second highest maximal bid. 47 Second-Price Auction Game Players: n bidders. Bidder is valuation is vi, we order v1> > vn > 0, without loss of generality. Strategies: bidder is maximal bid is bi.

Let bi = max {bj : j different from i}. Payoffs: ui(b1, ,bn) = vi - bi 0 if bi < bi if bi > bi 48 Nash Equilibria 1. (b*1, , b*n) = (v1, , vn). Bidder 1 wins the object, payoff: v1 b*2 = v1 v2 > 0. If bidding b1 < v2, she loses the object, the payoff is 0.

If bidding b1 > v2, her payoff is v1 v2 > 0. The payoff of bidders i = 2, , n is 0. If bidding bi > v1, the payoff is vi b1 = vi v1 49 2. (b*1, , b*n) = (v1, 0, , 0) Bidder 1 wins the object, her payoff is v1. The payoff of bidders i = 2, , n is 0. If bidding bi > v1, the payoff is vi b1 = vi v1 < 0. If bidding bi < v1, she loses the object, the payoff is 0. 3. (b*1, , b*n) = (v2, v1 , 0, , 0) Bidder 2 wins the object, payoff v2 b1 = v2 v2 = 0. To win, bidder i = 1, 3, , n must bid bi

50 Weakly Dominant Solution The Nash Equilibrium (b*1, , b*n) = (v1, , vn) is the unique weakly dominant solution. bi < bi < vi bi < bi or or bi > vi bi = bi & i bi=bi & i wins vi - bi 0 0 bi < v i

loses bi = v i vi - bi vi - bi 0 51 bi = vi vi - bi vi < bi < bi bi > bi or or bi =bi & i bi =bi & i loses 0

0 wins bi > vi vi - bi vi bi (< 0) bi < vi 0 In sum, bidding bi = vi yields at least as high a payoff as bidding bi > vi or bi < vi for any opponents bids. 52 Summary of the Lecture

Definition of Games in Strategic Form. Definition of Solution Concepts Nash Equilibrium, Dominance and Rationalizability. Applications Cournot Oligopoly, Bertrand Duopoly, Downsian Electoral Competition, Vickrey Second Price Auction. 53 Preview Next Lecture

Mixed Strategies. Nash Equilibrium and Rationalizability. Correlated Equilibrium. 54

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