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An in-depth analysis of subgame perfection in extensive form games. It explains the concept of selten's subgame perfect equilibrium, how to find it using the one-step deviation principle, and its application to both finite and infinite horizon games. The document also covers infinitely repeated games and the grim strategy.
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+Definition of a Game Γ Set of players Order of moves (game tree) Set of action at each node information sets Payoffs Notation: Hi is the set of player’s i information sets. A(hi) is the set of player’s i actions at hi. Ai = ∪hi A(hi) De Ai fiwithnition: A pure strategy for player si(hi) ∈ A(hi) for all hi ∈ Hi (^) iis a map. si : Hi →
The set of strategies for player i is Si. Si = ×hi A(hi) Mixed strategy is a randomization over Si.
Example: Entry game
E
-1, - In I 1, 1
F A Out 0, 4 Pure Strategies: SE = {In, Out} and SI = {F, A}. Normal Form representation:
In -1, -1F^ 1, 1A Out 0, 4 0, 4 NE: (Out, F), (In, A) and (out, α ∈ h 1 αF + (1 − α)A) with 2 ,^1 i THP: (In, A). The other equilibria are based on an emptythreat.
To eliminate NE with non-credible actions with use Selten’ssubgame perfect equilibrium concept.
Definition: σ is a SGPE if it is a NE in every subgame. (look at previous examples)
How do we find SGPE?
+Instep deviation condition (OSDC). finite horizon games by backwards induction or the one
+In infinite horizon games by the OSDC.
To describe OSDC we focus on multi-stage games withperfect information.
history: ht^ = (a^1 , a^2 , ..., a t−^1 )
sfromi satis s fies OSDC if there does not exist any s^0 i different u(s^0 i^ in only one^ ht^ and is the same everywhere else and i /ht)^ > u(si^ /ht).
+for inuous at infinite horizon extensive games: For games contin-finity ( sup h,h^0 s.t.ht=h^0 t
¯¯Ui(h) − Ui(h (^0) )¯¯ (^) → 0 as tis (^) fi→ ∞es the OSDC. for all i as ) σ is a SGPE if and only if it sat-
Proof (idea): =⇒ trivial
SGPE. Thus,^ ⇐=^ Suppose not: ∃σb σ^ satisfies the OSDC but is not ht. i^ s.t.^ Ui(σb^ i)^ > Ui(σ^ i)^ at some subgame Ui(σb (^) i) > Ui(σ (^) i) + 2ε Let σ^0 i be equal to σb (^) i for t < T and equal to σ (^) i for t ≥ T. By continuity at infinity, for T large: Ui(σ^0 i) > Ui(σb (^) i) − ε Then, Ui(σ^0 i) > Ui(σ (^) i) + ε By one step deviation principle for U finite extensive games:
Example:
C 1, 1C^ -1, 2D D 2, -1 0, 0 t=0,1.....
Ui = (^) tP∞=0 δ t^ gi(ati , at −i)
Two interpretations of δ:
Infinitely Repeated Games: +When is (Grim, Grim) a SGPE? *No D before: Ui(C, Grim) = 1 + δ + δ^2 + .... = (^1) −^1 δ Ui(D, Grim) = 2 + 0δ + 0δ^2 + .... = 2 Then, Ui(C, Grim) ≥ Ui(D, Grim) if δ ≥ 12.
*D before: Ui(D, Grim) = 0 Ui(C, Grim) = − 1 Therefore, cooperation can be supported if δ ≥ 12.
+Player 1 makes a division and player 2 can accept or not +"Cake" of size 1. +Payoffs If player 1 offers (x, 1 − x) and player 2 accepts: U 1 =^ x^ and^ U 2 = 1^ −^ x. If player 2 rejects: U 1 , 2 = 0.
Proposition: Any divisionNE outcome. (a, 1 − a) with a ∈ [0, 1] is a
Proof: s 1 = a
s 2 =
( A if x ≤ a R otherwise Check that s 1 and s 2 are optimal at t = 1. *Player 1: If offer x < a, then A and U 1 < a If offer x = a, then A and U 1 = a If offer x > a, then R and U 1 = 0 *Player 2: Given that player 1 o does not matter and sffers a, what player 2 does for x 6 = a
Proposition: The unique SGPE outcome is (1, 0). Proof: By backwards induction we have that in a SGPE player 2must play
s 2 =
( A if x < 1 R if x = 1 or^ s^02 =^ A^ for all^ x. If player 2 chooses s^02 , then player 1 would choose x = 1. If player 2 chooses s 2 , then player 1 has no best response.
+Results: Market behavior: a) Converged quickly to SGPE. b) No differences across countries. Bargaining behavior: a) Significantly different from SGPE. (inefficient) b) Substantial differences across countries.
tions do not).c) Proposals maximize expected value.^ (Rejec-
versely correlated to size of od) Within every country probability of R is in-ffer.
perience. e) Differences across countries increased with ex-
