Lecture III: Normal Form Games Recommended Reading: Dixit & Skeath: Chapters 4, 5, 7, 8 Gibbons: Chapter 1 Osborne: Chapters 2-4

Recap & Introduction A Game: –Players + Strategy Set (rules & plans of action) + Outcomes (payoffs) Nash Equilibrium: –A best response to a best response –i.e., no player wants to alter strategy unilaterally –If G = {S 1,…, S n ; u 1,…, u n }, the strategies (s* 1,…,s* n ) are a Nash equilibrium if  i u i (s* 1,…,s* i-1, s i *, s* i+1,…, s* n ) ≥ u i (s* 1,…,s* i-1, s i, s* i+1,…, s* n )

Prisoners’ Dilemma Player 2 ConfessSilent Player 1Confess2, 24, 0 Silent0, 43, 3 Normal form representation Players have discrete strategies, confess, stay silent Cells contain payoffs, row’s first, column’s second

Prisoners’ Dilemma Player 2 ConfessSilent Player 1Confess2, 24, 0 Silent0, 43, 3 Identify NE by eliminating strictly dominated strategies For i, a strategy, s i ´, is strictly dominated by s i ´´ if: u i (s 1...s i-1, s i ´, s i+1...s n ) < u i (s 1...s i-1, s i ´´, s i+1...s n )  (s 1...s i-1, s i, s i+1...s n )  S i.e., i always does better not playing s i ´ irrespective of other players’ strategies

Prisoners’ Dilemma Player 2 ConfessSilent Player 1Confess2, 24, 0 Silent0, 43, 3 Conversely, a strictly dominant strategy maximizes u i irrespective of what others do i should never play a strictly dominated strategy If a strictly dominant strategy exists, i should play it. But NE does not hinge on existence of strictly dominant strategies

Prisoners’ Dilemma Player 2 ConfessSilent Player 1Confess 2, 2 4, 0 Silent 0, 4 3, 3 Player i’s Logic: 1.Take j’s strategy as fixed 2.Compare payoffs under different strategies Given s j = “Silent”: i.u i (s i (C), s j (S)) = 4 ii.u i (s i (S), s j (S)) = 3 Given s j = “Confess”: iii.u i (s i (C), s j (C)) = 2 iv.u i (s i (S), s j (C)) = 0

Prisoners’ Dilemma Player 2 ConfessSilent Player 1Confess 2, 24, 0 Silent 0, 43, 3 Player i’s Logic: 1.Take j’s strategy as fixed 2.Compare payoffs under different strategies Given s j = “Silent”: i.u i (s i (C), s j (S)) = 4 ii.u i (s i (S), s j (S)) = 3 Given s j = “Confess”: iii.u i (s i (C), s j (C)) = 2 iv.u i (s i (S), s j (C)) = 0

Prisoners’ Dilemma Player 2 ConfessSilent Player 1Confess 2, 2 4, 0 Silent0, 43, 3 Player i’s Logic: 1.Take j’s strategy as fixed 2.Compare payoffs under different strategies Given s j = “Silent”: i.u i (s i (C), s j (S)) = 4 ii.u i (s i (S), s j (S)) = 3 Given s j = “Confess”: iii.u i (s i (C), s j (C)) = 2 iv.u i (s i (S), s j (C)) = 0

Weak Dominance Some games do not have strictly dominant strategies Battle of Bismark Sea: –US has no dominant strategy –For Japan, N weakly dominates S, (i.e., N at least as good as S no matter what US does, and sometimes better than S) –This allows US to choose strategy & generates NE Japan NS USN 2,-2 S1,-13,-3

Weak Dominance Elimination of weakly dominated strategies not sufficient to identify all NE. –US & Canada run on 110V (convenient) –Both switching to 220V brings world convergence & extra convenience –If only one switches, world convenience offset by continental inconvenience –For US & Canada: U(220V)  U(110V) irrespective of other’s strategy –{220V, 220V} is NE... –{110V, 110V} also NE: If US is 110V would Canada unilaterally switch to 220V? Canada 110V220V US110V0, 0 220V0,01, 1 Canada 110V220V US110V0, 0 220V0,01, 1

Multiple Equilibria Not all games have unique NE Consider the following games: A.Pure Coordination B.Battle of the Sexes C.Stag Hunt CCanada FP USF2,20, 1 P1, 01, 1 BChris CT C2, 30, 0 T 3, 2 ACar 2 LR Car 1 L2,20,0 R 2,2 Carla

Multiple Equilibria Not all games have unique NE Consider the following games: A.Pure Coordination –no reason to for i to resist {L, L} over {R, R} –but no compelling reason for i to play L or R CCanada FP USF2,20, 1 P1, 01, 1 BChris CT C2, 30, 0 T 3, 2 ACar 2 LR Car 1 L2,20,0 R 2,2 Carla

Multiple Equilibria Not all games have unique NE Consider the following games: B.Battle of the Sexes –conflict & coordination –still no reason to choose {C, C} over {T, T} –Credible commitment? CCanada FP USF2,20, 1 P1, 01, 1 BChris CT C2, 30, 0 T 3, 2 ACar 2 LR Car 1 L2,20,0 R 2,2 Carla

Multiple Equilibria Not all games have unique NE Consider the following games: C.Stag Hunt –Pareto optimality provides compelling reason for {F, F} –but {P, P} remains NE – despite P being weakly dominated by F CCanada FP USF2,20, 1 P1, 01, 1 BChris CT C2, 30, 0 T 3, 2 ACar 2 LR Car 1 L2,20,0 R 2,2 Carla

No Pure Strategy Nash Equilibria Some games do not have pure strategy NE Consider reformulation of Battle of Bismark Sea game –Neither player has a dominant strategy –At every cell, at least one player want to alter strategy –No NE! Japan NS USN2,24,1 S5,01,4

Mixed Strategies If i has k = 1…K pure strategies, S i = {s i1,…,s iK } in G = {S 1,…, S n ; u 1,…, u n }, then a mixed strategy is a probability distribution, p i = (p i1,…,p iK ) s.t. 0 ≤ p ik ≤ 1 and  p ik = 1 If G = {S 1,…,S n ; u 1,…,u n }, where n is finite and S i is finite  i   at least one NE, possibly involving mixed strategies Informally, i plays all her available pure strategies with some probability (perhaps Pr = 0 for some) Interpret i’s mixed stratgy as j’s uncertainty about what strategy i will actually adopt

Mixed Strategies in Practice US plays N with Pr = p; Japan plays N with Pr = q US: –EU(N) = 4-2q –EU(S) = 1+ 4q –EU(N) > EU(S) iff ½ > q Japan NqNq S (1-q) US N p2,24,14-2q S (1-p)5,01,41+4q 2p2p4-3p

Mixed Strategies in Practice US plays N with Pr = p; Japan plays N with Pr = q Japan: –EU(N) = 2p –EU(S) = 4 – 3p –EU(N) > EU(S) iff p > 4/5 Japan NqNq S (1-q) US N p2,24,14-2q S (1-p)5,01,41+4q 2p2p4-3p

Best-Response Curves p /5 1/2 q If q < 1/2, US should play N with Pr(p) = 1 … (it gets more utility) If p >4/5, Japan should play N with Pr(q) = 1… (ditto) US’s best-response curve Japan’s best-response curve

Continuous Strategies: The Cournot Game Two firms in competition, i & j q i and q j denote quantities of single, homogenous good produced by each firm P(Q) = a – Q is market clearing price, where Q = q i + q j. –N.B. (If a Q.) C i (q i ) = cq i, i.e., constant costs per unit

Continuous Strategies: The Cournot Game Each firm’s strategy space is S i = [0,  ) so any q i ≥ 0 is admissible (though a puts an implicit limit on q i ). Each firms’ payoffs are equal to their revenues, i.e., market price  quantity – costs:  i (q i, qj) = q i [P(q i + q j ) – c] = q i [a – (q i + q j ) – c] Firms choose quantities simultaneously; how much does / should each produce?

Continuous Strategies: The Cournot Game Each firm’s faces an optimization problem: max  i(q i, q j *) = max q i [a – (q i + q j *) – c] 0≤ q ≤  To solve, we need to obtain i’s first-order condition, i.e., differentiating above w.r.t q i, set equal to 0, and solve: q i * = ½(a – q j * – c) Game is symmetric in strategies & payoffs, so: q j * = ½(a – q i * – c)

Continuous Strategies: The Cournot Game Each firm’s wants to produce: q i * = ½(a – q j * – c)[1] q j * = ½(a – q i * – c)[2] Equations 1 & 2 tells us what q i * and q j * are, so substitute q j * from 2 into Equation 1 and solve: qi* = ½(a – ½(a – q i * – c) – c) = (a + c)/3 Same holds for j. The NE = q i * = q j * = (a + c)/3

Best-Response Curves qiqi (a + c)/3 0 qjqj Drawing best response curves helps to get a better sense of the NE The NE occurs at the intersection of each firm’s payoff (i.e., revenue) curve. (a + c)/3 q i [a-(q i +q j )-c]

Combining Discrete & Continuous Strategies Lichbach (1990) provides examples of games in which players’ strategies are discrete but their payoffs are continuous Just replace the 1s, 2s & 3s etc in our earlier examples with a payoff function defined by variables as in the Cournot game e.g., if for Row B > D > A > C and for Column C > D > A > D, then the game has the form of a Prisoner’s Dilemma Column III RowIAB IICD

Combining Discrete & Continuous Strategies In contrast, if for Row A = D > B = C, & for Column, A = D > B = C, then the game is a pure coordination game We can replace A, B, C, & D with functions if we wished. Proving that {I, I} is a unique NE then requires showing conditions under which: 1.uR(sR(I), sC(.)) > uR(sR(II), sC(.)) & 2.uR(sR(I), sC(.)) > uR(sR(II), sC(.)) i.e., that for Row & Column, II is strictly dominated by I Column III RowIAB IICD