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UADPhilEcon Ec10 Strategic Choices Lecture 3 The Refinement Project continued Dynamic Games Two dynamic games 20 matches on the table. Two players take turns to collect one or two at a time. The one collecting the 20 th wins!
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UADPhilEcon Ec10Strategic Choices Lecture3 The Refinement Project continued Dynamic Games
Two dynamic games • 20 matches on the table. Two players take turns to collect one or two at a time. The one collecting the 20th wins! • 20 golden coins on the table. Aim: To get as much as possible for yourself. Two players take turns to collect one or two at a time. If a player collects two, game over!
Two types of backward induction • Backward induction without CKR • Backward induction with CKR or Nash backward induction NBI equivalent to CAB
Restricting the number of equilibria by introducing sequence/time
Subgame perfect Nash equilibrium (SPNE) – Selten, 1975 • A set of strategies for all players and for each node of the game (also known as a strategy profile) such that: (a) player i’s strategy is a best reply to the strategies of the others at any point of the game after which it is i’s turn to play, and (b) this is so independently of what has happened beforehand.
Sequential Equilibrium (Kreps and Wilson, 1982) • The sequential equilibrium comprises both a strategy profile and a belief system, where the latter specifies for each information set of each player a belief held by the player who is about to act at that information set regarding what has happened in the past (or, equivalently, which precise node she/he at).
The Short Centipede An alternative, non-equilibrium, logic Node 2: [p½2000 + p½800 + (1-p)800] > 1000 That is, p is at least equal to ⅓. Node 1: Worth a bluff if p will exceed ⅓ with prob 1 in 2000!
Harsanyi’sdefence (based on Kreps and Wilson’s sequential eqlm) Stage 3: Pr(a3|R is rational) = 0 Pr(d3|R is rational) = 1 Pr(a3|R is irrational) = ½ Pr(d3|R is irrational) = ½ Stage 2: ERC(a2) = 2000[½ p2] + 800[½ p2 + (1-p2)] So, for C to lean toward a2 ERC(a2) must exceed the sure 1000 pay-off from d2: i.e. 2000[½ p2] + 800[½ p2 + (1-p2)] ≥ 1000 or, p2 ≥ ⅓. Stage 1: q=Pr(p2 ≥ ⅓| a1). If q is large enough, this means that R feels that the prospects of a successful bluff are good.
Imposing equlibrium… In sequential equilibrium: p2=⅓. Why? Suppose p2>⅓ This is tantamount to saying that R managed to force C to play pure strategy ‘across’ (a2) at stage 2 of the game. But how can this be? And vice versa
Thus,… Or r=2p1
This game’s sequential equilibrium: • p1≥⅓, in which case at Stage 1 a rational R always bluffs (i.e. plays pure strategy a1), and an irrational R randomises between a1and a2; at Stage 2 C always plays a2 (as long as he gets a chance), and at Stage 3 a rational R plays d3 whereas an irrational R randomises between d3 and a3. • p1<⅓, in which case at Stage 1 a rational R bluffs (i.e. plays a1) with probability r=2p1, while an irrational R randomises between a1 and a2; at Stage 2 C randomises between a2 and d2 (as long as he gets a chance), and at Stage 3 a rational R plays d3 whereas an irrational R randomises between d3 and a3.
BUT… • The relaxation of CKR was bought at a hefty price: • Another, more toxic, type of common knowledge: • the common knowledge of R’s initial reputation (p1) as well as the deduced probability with which a rational R will bluff (r)
