Beer-Quiche Game (Cho and Kreps, QJE 102(2):183 (1987))
"Strategic normalization of extensive games versus the revelation principle" by Roger Myerson Celebration of CME-MSRI Prize for Robert Wilson, 2 Feb 2017
When I first heard Robert Wilson talk about sequential equilibria of extensive-form games, I thought that equilibrium concepts should be defined through the normal representation in strategic form (with refinements like perfectness or properness). But he and I switched at some point after 1982, and now I see fundamental reasons to think that the normal representation might not be enough. The concept of strategic normalization allowed game theorists to see one-stage games where players make simultaneous strategic choices (the strategic form) as a general model that can subsume all multi-stage games (in extensive form). The analysis of games can also be simplified by admitting communication possibilities in the solution concept, instead of in the game model, using the revelation principle. But admitting communication in this way invalidates the principle of strategic normalization. We consider two examples to illustrate this point. (Admitting small chance of mediated communication yields simple refinement: Nash without sequentially codominated actions; my Econometrica 1986. W=r?) 1 An example given in extensive form (Figure 6.3, Myerson, Game Theory). Mon TueAM TuePM MedPlan: εPlan:
(0) (1-ε)
(0.5) (ε/2)
(0) (0)
(0) (0)
(0.5) (ε/2)
2's strategy 1's strategy x2 y2 a1x1 2,2 2,2 a1y1 2,2 2,2 b1x1 5,1 0,0 b1y1 0,0 1,5 dominated by a1 (then y2 dom'd)
Mediation plan: announce "x" or "y" (each with pr=0.5) on TuesdayAM. ε-mediated plan: ...mediator exists with probability ε, "y"=silence. 2 The Beer-Quiche game (Cho and Kreps, QJE 102(2):183 (1987)).
2beer/2quiche 1strong/1weak d/d d/n n/d n/n b/b 0.9, 0.1 0.9, 0.1 2.9, 0.9 2.9, 0.9 b/q 1.0, 0.1 1.2, 0.0 2.8, 1.0 3.0, 0.9 dom4 by b/b q/b 0.0, 0.1 1.8, 1.0 0.2, 0.0 2.0, 0.9 dom1 by .8[q/q]+.2[b/q] q/q 0.1, 0.1 2.1, 0.9 0.1, 0.1 2.1, 0.9 dom3 by b/q dom1 by n/n dom2 by n/n dom3 by n/d .8[q/q]+.2[b/q] 0.28 1.92 0.64 2.28
3 The "counterintuitive" quiche equilibrium in Beer-Quiche can be made strict with a small probability of mediation.
A mediator (M) exists with small probability ε. If M exists, M asks 1 his type. If "strong," M recommends "beer" to 1, "duel iff quiche" to 2. If "weak," with proby 0.5(1-δ), M recommends "beer" to 1, "duel iff quiche" to 2; with proby 0.5δ, M recommends "beer" to 1, no communication with 2; with proby 0.5(1-δ), M recommends "quiche" to 1, no communication with 2; with proby 0.5δ, M recommends "quiche" to 1, "duel iff quiche" to 2. Equilibrium: honest-&-obedient with mediator, "quiche" & "duel iff beer" without.
2 hears mediator "duel iff quiche" 2 does not hear mediator beer quiche beer quiche 1 strong 0.9ε 000.9(1-ε) 1 weak 0.1ε0.5(1-δ) 0.1ε0.5δ 0.1ε0.5δ 0.1[ε0.5(1-δ)+(1-ε)] 2 prefers no duel duel duel no duel (ε<0.94) 1 prefers to obey mediator, or do quiche if none, given 2's expected behavior, δ<1/4. Strong type of 1 prefers honesty with mediator, and weak prefers honesty when δ<1/6. 4