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Quantal Response Equilibrium

Quantal Response Equilibrium. APEC 8205: Applied Game Theory Fall 2007. THE GAME. Players: N = {1,2,…,n} Strategies: S i = {s 1 i ,s 2 i ,…,s J(i) i } Strategy Profile: s = {s 1 ,s 2 ,…,s n } for s i  S i  i  N Strategy Space: S =  i  N S i

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Quantal Response Equilibrium

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  1. Quantal Response Equilibrium APEC 8205: Applied Game Theory Fall 2007

  2. THE GAME • Players: N = {1,2,…,n} • Strategies: Si = {s1i,s2i,…,sJ(i)i} • Strategy Profile: s = {s1,s2,…,sn} for si Si i  N • Strategy Space: S = iNSi • Individual Payoffs: ui(s)  • Everyone’s Payoffs: u(s) = {u1(s),u2(s),…,un(s)}

  3. SOME NOTATION • i: J(i) dimensional simplex •  = iNi • pi = (p1i,p2i,…,pJ(i)i) i • p = {p1, p2,…,pn}  • pi(si): Probability player i chooses strategy si • p(s) = i = 1npi(si): Probability of strategy profile s  S given p • Eui(p) = sS p(s)ui(s): Player i’s expected payoff

  4. DEFINITION p’ = {pi’, p-i’} is a Nash equilibrium if for all i  N and all pii, Eui(pi’, p-i’)  Eui(pi,p-i’) where p-i’ is p’ exclusive of pi’.

  5. MORE NOTATION • sji = {pi: pji = 1}: Player i’s pure strategy j • ji: Random error for player i and strategy j • Euji(p) = Eui(sji,p-i) + ji: i’s expected payoff for strategy j plus an error • i = (1i,2i,…,J(i)i): Collection of errors for player i • fi(i): Joint density of errors assuming E(i) = 0 • fji(ji): Marginal density of error •  = (1,2,…,n): Errors for all players • f = (f1,f2,…,fn): Joint densities of errors for all players

  6. ASSUMPTION Player i chooses the strategy j when Euji(p)  Euki(p)  k = 1,2,…,J(i).

  7. IMPORTANT NOTES • Player iknows i, but not k for i  k. • Player i only knows the distribution fk(k), which means k’s strategy choice is random from the perspective of i. • k’s strategy choice is not uniformly random because it also depends on his payoffs and lack of knowledge of j’s specific strategy choice.

  8. BACK TO NOTATION • Rji(p) = {i| Euji(p)  Euki(p)  k = 1,2,…,J(i)} • Region of errors that make strategy j optimal for player i • ji(p) = • probability strategy j optimal for player i

  9. ANOTHER DEFINITION •  is a Quantal Response Equilibrium if ji = ji()  j = 1,2,…,J(i) and i  N where ji is the probability player i chooses strategy j and  = {1, 2,…, n} where i = {1i, 2i,…, J(i)i}  i  N.

  10. COMMENT Assuming the ji’s are identically and independently distributed (iid) extreme value (Weibull), the QRE implies the logit function ji = where  is a parameter that is inversely related to the dispersion or variance of error. For  = 0, probabilities are uniform. As  approaches , the dispersion of error approaches 0 and the QRE approaches a Nash equilibrium.

  11. GENERAL EXAMPLE If 1i = i and 2i = 1 - i for i = 1,2, the QRE will solve and

  12. MORE SPECIFIC EXAMPLE

  13. ANOTHER SPECIFIC EXAMPLE • NE = {(0.50, 0.50), (0.50, 0.50)} • Observed: • Goeree & Holt = {(0.48, 0.52), (0.48, 0.52)} • Our Class = {(0.75,0.25),(0.44,0.56)}

  14. ANOTHER SPECIFIC EXAMPLE • NE = {(0.50, 0.50), (0.125, 0.875)} • Observed: • Goeree & Holt = {(0.96, 0.04), (0.16, 0.84)} • Our Class = {(0.57,0.43),(0.20,0.80)}

  15. ANOTHER SPECIFIC EXAMPLE • NE = {(0.50, 0.50), (0.909, 0.091)} • Observed: • Goeree & Holt = {(0.08, 0.92), (0.80, 0.20)} • Our Class = {(0.29,0.71),(0.70,0.30)}

  16. PLAYER 1’s QRE

  17. PLAYER 2’s QRE

  18. EMPIRCAL ANALYSIS • N pairs of subjects. • yi{(Top, Left), (Top, Right), (Bottom, Left), (Bottom, Right)} • Let y = {y1,…,yN}, so the probability of y is L = i=1NPr(yi) where • Solve for : 1() and 2(). • Optimize L for  0.

  19. WHAT HAVE OTHERS DONE EMPIRICALLY • McKelvey and Palfrey: Had subjects play a variety of games. • Hypotheses: • Random Play • Rejected • Nash Play • Rejected • Learning (e.g.  decreases with experience) • Results Mixed

  20. GOOD EXERCISE • Data From Class Room Experiment Last Year • Treatment 1: {(T, L), (T, R), (B, L), (B, R)} = {3,2,1,1} • Treatment 2: {(T, L), (T, R), (B, L), (B, R)} = {0,6,1,0} • Treatment 3: {(T, L), (T, R), (B, L), (B, R)} = {2,0,3,2} • Questions: • Is play strictly random? • Does  differ across treatments? • What are the estimated probabilities of Top & Bottom and Left & Right for each treatment?

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