Extensive form games - contd

Proposition: Every finite game of perfect information Γ E has a pure-strategy SPNE. Moreover, if no player has the same payoffs in any two terminal nodes, then there is a unique SPNE. Idea of sequential rationality can be extended to games with imperfect information. 1

Firm E Out In U(E) 0 U(I) 2 Firm E Fight Accommodate Firm I Fight Accomm Fight Accomm -3 1-2 3-1 -2-1 1 Firm E Firm I Accommodat Fight Accommodat 3,1-2,-1 Fight 1,-2-3,-1 I Accomm if E Fight if E plays In plays In Out, Accomm if In 0,2 0,2 E Out, Fight if In 0,2 0,2 In, Accomm if In 3,1-2,-1 In, Fight if In 1,-2-3,-1 2

Normal form version has 3 pure strategy NE: ((out, accomm if in),(fight if firm E plays in)); ((out, fight if in),(fight if firm E plays in)); ((in, accomm if in),(accomm if firm E plays in)) However (accomm, accomm) is the sole NE in the simultaneous move game that follows entry. Logic of sequential rationality implies that only the last equilibrium is a SPNE. 3

Above game had unique NE in post-entry subgame. In such games, the SPNE concept rules out history dependence of strategies. With more than 1 NE in post-entry game, behaviour earlier could depend on which equilibrium resulted after entry. 4

Firm E Out In U(E) 0 U(I) 2 Firm E Small niche Large niche Firm I Small Large Small Large niche niche niche niche -6-1 1-3 -6 1-1 -3 Firm E Firm I Small Large Small -6,-6-1,1 Large 1,-1-3,-3 E E Out In Out In 0 1 0-1 2-1 2 1 5

Two pure strategy NE in post-entry game: (large niche, small niche); (small niche, large niche) Suppose that firms will play (large niche, small niche). Then entrant optimally chooses to enter. So one SPNE is (σ E, σ I ) =((in, large niche if in),(small niche if firm E plays in)). Other SPNE is (σ E, σ I ) =((out, small niche if in),(large niche if firm E plays in)). 6

Beliefs and sequential rationality Firm E Out In1 In2 Firm I U(E) 0 U(I) 2 Fight Accomm Fight Accomm -1 3-1 2-1 0-1 1 Fight if Accomm if entry occurs entry occurs Out 0,2 0,2 In1-1,-1 3,0 In2-1,-1 2,1 7

Two pure-strategy NE in above game: (out, fight if entry occurs), (in1, accomm if entry occurs) But again, (out, fight if entry occurs) does not seem reasonable; regardless of type of entry (In1, In2), incumbent prefers to accommodate once entry occurs. But criterion of subgame perfection cannot be applied here to eliminate non-sensible equilibrium, since only subgame is entire game! 8

New equilibrium concept - weak perfect Bayesian equilibrium (weak PBE) or weak sequential equilibrium Requires that at any point in game, a player s strategy prescribe optimal actions from that point on, given her opponents strategies and her beliefs about what has happened so far, and that her beliefs be consistent with the strategies being played. 9

Definition: A system of beliefs µ in extensive form game Γ E is a specification of a probability µ(x) [0, 1] for each decision node x in Γ E such that Σ x H µ(x) = 1 for all information sets H. 10

Definition: A strategy profile σ = (σ 1,.., σ I ) in extensive form Γ E is sequentially rational at information set H given a system of beliefs µ if, denoting by ι(µ) the player who moves at information set H, we have E[u ι(h) H, µ, σ ι(h), σ ι(h) ] E[u ι(h) H, µ, σ ι(h), σ ι(h) ] for all σ ι(h) (S ι(h) ). If strategy profile σ satisfies this condition for all information sets H, then we say that σ is sequentially rational given belief system µ. 11

Suppose each player s equilibrium strategy assigns a strictly positive probability to each possible action at every one of her information sets. i.e., every information set is reached with positive probability. For each node x H i, i should compute probability of reaching that node given play of strategies σ, P r(x σ) and she should assign conditional probabilities using Baye s rule: P r(x H, σ) = P r(x σ) Σ x H P r(x σ) 12

Suppose in game above, firm E is using the completely mixed strategy such that P r( out ) = 1 4, P r( In1 ) = 1 2 and P r( In2 ) = 1 4. Probability of reaching firm I s information set given this strategy is 3 4. Using Baye s rule, probability of being at the left node of I s information set conditional on reaching it is 2 3 and that of being at the right node is 1 3. 13

Pr(reaching information set)=pr(not out )=1-1 4 = 3 4 Pr(L/not out )= P r(l).p r(not out L) P r(l).p r(not out L)+P r(r).p r(not out R) = (1/2).1 (1/2).1+((1/4).1 = 2 3 Pr(L/not out )=1 2 3 = 1 3 14

When players are not playing completely mixed strategies, some information sets may not be reached with positive probability. We then cannot use Baye s rule to compute conditional probabilities. In such cases, the weak PBE concept allows us to assign any beliefs at these information sets (hence the term weak ). 15

Definition: A profile of strategies and system of beliefs (σ, µ) is a weak PBE in extensive form game Γ E if it has the following properties: (i) The strategy profile σ is sequentially rational given belief system µ; (ii) The system of beliefs µ is derived through Bayes rule whenever possible. 16

Player 1 Out 2 2 In Player 1 U D p Player 2 (1-p) L R L R 3 0 0 1 1 0 0 3 Player 2's optimal strategy is to play L for p>(2/3) But only p=1 is sequentially rational 17

One shortcoming of the weak PBE concept is that it places no restrictions on beliefs off the equilibrium path. Because of this, a weak PBE need not be subgame perfect. Lets go back to the 1st game we looked at. 18

A weak PBE of this game is (σ E, σ I )=((out, accommodate if in),(fight if firm E plays in )) with beliefs for firm I that assign probability 1 to firm E having played fight. But these strategies are not subgame perfect; they are not a NE in the post-entry subgame. Problem arises because firm I s post-entry belief about firm E s post-entry play is unrestricted by the weak PBE concept - firm I s information set is off the equilibrium path. 20