Copyright (C) 2013 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of the Creative

Transcription

1 Copyright (C) 2013 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of the Creative Commons attribution license

2 Long Run versus Short Run Player a fixed simultaneous move stage game Player 1 is long-run with discount factor δ actions a A a finite set utility u ( a, a ) Player 2 is short-run with discount factor 0 actions a utility u A a finite set 2 2 ( a, a )

3 What it is about the short-run player may be viewed as a kind of representative of many small long-run players the usual case in macroeconomic/political economy models the long run player is the government the short-run player is a representative individual 2

4 Example 1: Peasant-Dictator (1,2) (3,0) Low High (0,1) 1 Eat Grow 2 3

5 Example 2: Backus-Driffil Low High Low 0,0-2,-1 High 1,-1-1,0 Inflation Game: LR=government, SR=consumers consumer preferences are whether or not they guess right Low High Low 0,0 0,-1 High -1,-1-1,0 with a hard-nosed government 4

6 Repeated Game history ht = ( a1, a2,, at ) null history h 0 i behavior strategies α t i = σ ( h t 1) long run player preferences average discounted utility T t = 1 1 ( 1 δ) δ t u i ( at ) note that average present value of 1 unit of utility per period is 1 5

7 Equilibrium Nash equilibrium: usual definition cannot gain by deviating Subgame perfect equilibrium: usual definition, Nash after each history Observation: the repeated static equilibrium of the stage game is a subgame perfect equilibrium of the finitely or infinitely repeated game strategies: play the static equilibrium strategy no matter what 6

8 perfect equilibrium with public randomization may use a public randomization device at the beginning of each period to pick an equilibrium key implication: set of equilibrium payoffs is convex 7

9 Example: Peasant-Dictator (1,2) (3,0) Low High (0,1) 1 Eat Grow 2 normal form: unique Nash equilibrium high, eat eat grow low 0*,1 1,2* high 0*,1* 3*,0 8

10 Static Benchmarks payoff at static Nash equilibrium to LR player: 0 precommitment or Stackelberg equilibrium precommit to low get 1 mixed precommitment to get 2 minmax payoff to LR player: 0 9

11 Payoff Space utility to long-run player mixed precommitment/stackelberg = 2 best dynamic equilibrium =? pure precommitment/stackelberg = 1 Set of dynamic equilibria static Nash = 0 worst dynamic equilibrium =? minmax = 0 10

12 Repeated Peasant-Dictator finitely repeated game final period: high, eat, so same in every period Do you believe this?? Infinitely repeated game begin by low, grow if low, grow has been played in every previous period then play low, grow otherwise play high, eat (reversion to static Nash) claim: this is subgame perfect 11

13 When is this an equilibrium? clearly a Nash equilibrium following a history with high or eat SR play is clearly optimal for LR player may high and get (1 δ)3+ δ0 or low and get 1 so condition for subgame perfection (1 δ)3 1 δ 2/3 12

14 Equilibrium Utility equilibrium utility for LR 1 0 δ 2/3 1 13

15 General Deterministic Case Fudenberg, Kreps and Maskin max u 1 ( a) mixed precommitment/stackelberg v 1 best dynamic equilibrium pure precommitment/stackelberg Set of dynamic equilibria static Nash v 1 worst dynamic equilibrium minmax min u 1 ( a) 14

16 Characterization of Equilibrium Payoff 1 2 α = ( α, α ) where α 2 is a b.r. to α 1 α represent play in the first period of the equilibrium 1 1 w ( a ) represents the equilibrium payoff beginning in the next period v ( 1 δ ) u ( a, α ) + δw ( a ) v = ( 1 δ ) u ( a, α ) + δw ( a ), α ( a ) > 0 v w ( a ) v

17 Simplified Approach impose stronger constraint using n static Nash payoff for best equilibrium n w( a) v for worst equilibrium v w( a) n avoids problem of best depending on worst remark: if we have static Nash = minmax then no computation is neede for the worst, and the best calculation is exact. 16

18 1 2 fix α = ( α, α ) max problem where α 2 is a b.r. to α v ( 1 δ) u ( a, α ) + δw ( a ) v = ( 1 δ ) u ( a, α ) + δw ( a ), α ( a ) > 0 n w ( a ) v how big can w ( a ) be in = case? Biggest when u ( a, α ) is smallest, in which case w ( a ) = v v = ( 1 δ ) u ( a, α ) + δv

19 Summary conclusion for fixed α min 1 1 u ( a, α ) a α ( a ) > 0 i.e. worst in support v 1 = α α BR ( α ) a α ( a ) > 0 max min u ( a, ) observe: mixed precommitment v 1 pure precommitment 18

20 Peasant-Dictator Example eat grow low 0*,1 1,2* high 0*,1* 3*,0 p(low) BR worst in support 1 grow 1 ½<p<1 grow 1 p=1/2 any mixture 1 (low) 0<p<½ eat 0 p=0 eat 0 19

21 Check the constraints 1 1 w ( a ) = v ( 1 δ ) u ( a, α ) δ n as δ 1 then w ( a ) v n 1 20

22 min problem fix α = ( α 1, α 2 ) where α 2 is a b.r. to α 1 v 1 ( 1 δ) u 1 ( a 1, α 2 ) + δw 1 ( a 1 ) v w ( a ) n Biggest u ( a, α ) must have smallest w ( a ) = v v = ( 1 δ ) u ( a, α ) + δ v conclusion v or v = max u ( a, α ) = BR min max u ( a, α ), that is, constrained minmax α ( α ) 21

23 Worst Equilibrium Example L M R U 0,-3 1,2 0,3 D 0,3* 2,2 0,0 static Nash gives 0 minmax gives 0 worst payoff in fact is 0 pure precommitment also 0 22

24 mixed precommitment p is probability of up to get more than 0 must get SR to play M 3p + ( 1 p) 3 2 and 3p 2 first one 3p + ( 1 p) 3 2 3p 3p 1 p 1/ 6 second one 3p 2 p 2 / 3 want to play D so take p = 1/ 6 get 1/ / 6 = 11/ 6 23

25 Utility to long-run player max u 1 ( a)=2 mixed precommitment/stackelberg=11/16 v 1 best dynamic equilibrium=1 pure precommitment/stackelberg=0 Set of dynamic equilibria static Nash=0 v 1 worst dynamic equilibrium=0 minmax=0 min u 1 ( a)=0 24

26 calculation of best dynamic equilibrium payoff p is probability of up p BR 2 worst in support <1/6 L 0 1/6<p<5/6 M 1 p>5/6 R 0 so best dynamic payoff is 1 25

27 Moral Hazard choose a observe y i A Y ρ( y a ) probability of outcome given action profile private history: h public history: h = ( a1, a2, ) i i i = ( y1, y2, ) strategy σ i ( h i, h) ( A i ) public strategies, perfect public equilibrium 26

28 Moral Hazard Example mechanism design problem each player is endowed with one unit of income players independently draw marginal utilities of income η { η, η} player 2 (SR) has observed marginal utility of income player 1 (LR) has unobserved marginal utility of income 27

29 Decisions, decisions player 2 decides whether or not to participate in an insurance scheme player 1 must either announce his true marginal utility or he may announce η independent of his true marginal utility non-participation: both players get γ = η + η 2 participation: the player with the higher marginal utility of income gets both units of income 28

30 normal form truth lie non-participation participate γ, γ η + γ η + γ, 2 2 γ, γ 3γ η, 2 2 p* = η makes player 2 indifferent γ 29

31 max u 1 ( a)= 3 γ 2 mixed precommitment/stackelberg= η + γ η η + ( 1 ) 2 γ 2 v 1 best dynamic equilibrium= η + γ 2 pure precommitment/stackelberg= η + γ 2 Set of dynamic equilibria static Nash=γ v 1 worst dynamic equilibrium=γ min u 1 ( a)=γ, minmax=γ 30

32 moral hazard case player 1 plays truth with probability p * or greater player 2 plays participate + F + H v = ( 1 δ ) η γ δ w( η) w( η) γ v ( 1 δ ) + δw( η) 2 v w( η), w( η) I K w( η ) must be as large as possible, so inequality must bind; w( η ) = v 31

33 3γ v = ( 1 δ ) + δw( η) 2 solve two equations v = η γ 2 v w η = ( 1 δ) 3γ / 2 ( ) δ Solving 32

34 Constraint check check that w( η) γ leads to δ F HG I K J η 2 2 γ from δ < 1 this implies η > 3η 33