Games

Question 1

IDSDS and NE

Answer 1

A nash equilibrium survives IDSDS

Question 2

IDSDS and NE unique

Answer 2

If IDSDS deletes all but one strategy profile, then it is NE

Question 3

Property of mixed strategy NE

Answer 3

• Expected payoff of all the pure strategies I play with non-zero prob are the same
• Expected payoff of all the pure strategies I play with non-zero prob is strictly greater than the expected payoff of all the pure strategies I play with prob 0

Question 4

Existence of equilibria

Answer 4

• Space of strategies are convex, non-empty and compact
• Payoff functions u_i of each player are continuous (s1,…,sn) and they are quasiconcave with respect s_i

Then there is a pure strategy NE

Every finite game has a NE, probably in mixed strategies.

Question 5

Def information set

Answer 5

Set of nodes I_i such that:

• Player i moves in all the sets of the node
• Player i has the same set of actions in each node of the set
• When the game reaches I_i, player i doesnt know at which of the nodes in I_i she is. She only knows that one the nodes of I_i has been reached.

Question 6

Strategy

Answer 6

a full contingent plan that specifies an action at each information set of the player

Question 7

zermello’s theorem

Answer 7

every finite game of perfect information has a pure strategy NE that can be derived through backward induction. If no player is indiferents between two actions at a particular node, then this NE is unique.

Question 8

Subgame definition

Answer 8

• It starts at a singleton n
• It contains all decision nodes that follow n and only those
• It does’t break any information set.

Question 9

Subgame perfect NE

Answer 9

player strategies constitute a NE at every subgame

Question 10

Existence of SPNE

Answer 10

every finite game has at least one NE, probably in mixed strategies

Question 11

Theorem

Answer 11

A finitely repeated game G with a unique NE at every stage has a unique SPNE that consists of T repetitions of the stage NE

Question 12

One period deviation principle for infinitely repeated games

Answer 12

To check wether (s1*,s2*) is a SPNE it is enough to check that there are no profitable “one information note” deviations

Question 13

Def feasible payoff

Answer 13

a payoff vector is feasible for the stage game if it is a convex linear conbination of the pure strategy payoffs of G.

Question 14

Folk theorem

Answer 14

Any payoff vector strictly greater to the one that is the result of the stage NE can be achieved as a result of a SPNE.

Question 15

Fundenberg-Maskin

Answer 15

any x>>_x where _x_i=min over s_-i { max over s_i of u_i(s_i, s_-i)