L 5 - 8, Ch 22, 24, 26, 27, 28 Flashcards
(30 cards)
Nature
In the extensive form, we model nature as “player 0,” whose decisions are made according to a fixed probability distribution. Because nature is not a strategic player, no payoff numbers are associated with nature. Graphically, nature’s decision nodes—also called chance nodes—are depicted by open circles, to distinguish them from the decision nodes of the strategic players. In games of incomplete information, rational play will require a player who knows his own type to think about what he would have done had he been another type, because even though player 1 knows his nature, player 2 does not
What matters in repeated interactions
reputation
Repeated game is
a game over several periods of time. It can be infinite or finite.
history at time t
Sequence of strategies of each player in each period up until t
The payoff of a finite t-period repeated game is
the sum of payoff earned in the t stage games
The payoff of an infinitely repeated game is
the discounted sum of payoffs earned in the stage games.
In finite games
each strategy combination of round 1 generates a new subgame, hence after round 1, there are as many subgames starting again as strategy combinations in the stage game. These subgames have the same structure as the stage game. The payoff matrix is the same as well. Hence, the subgames reached in period 2 has the same NE as the stage game.
The result is: In finitely repeated game, any sequence of stage game NE is a SPNE of the repeated game.
In finitely repeated games we can use credible threats and promises in period t=1 to support sequentially rational behavior in period t=0 that is not a NE of the stage game
Infinitely repeated games are used to model
long-term relationship. Discounting future payoffs allows us to model these situations. Discount factors can be denoted by delta
d= 1 means all future payoffs are valued equally
d= 0 means only today matters, and future payoffs have no value
0<=d<=1
Strategy in an infinitely repeated game may be very complex: we have to specify an action for a player at each IS (infinitely ISs), hence we have to specify an action for a player after every possible history a player could observe in the repeated game.
There are simple strategies as well. Consider a constant strategy that prescribes: always choose a. It describes that a player should choose a at each possible history, hence ignore the history and choose a. Another example is the trigger strategy.
Trigger strategy
specifies a history(ies) which trigger a change of behavior. Until a trigger history Is observed, player chooses one action, and after the trigger occurs, he chooses another action.
Trigger strategy consists of
A) Cooperative profile – prescription of sequence of actions that don’t form a stage game NE
B) Uncooperative profile – prescription to use stage game NE
C) Trigger from cooperative to uncooperative profile is triggered if any player deviates from the cooperative profile
TS can sustain cooperation in Prisoners dilemma as long as
as the players have high enough value for future cashflows and interact repeatedly. The TS is the SPNE
Folk Theorem
Consider Infinitely repeated game. Suppose the stage game has a NE that yields payoff w=(w1…w1). Let v=(v1…v1) be any feasible average per-period payoff such as vi>wi for any player i=1,..n. The payoff vector v can be supported as a SPNE outcome if delta is sufficiently close to 1.
This can be achieved by TSs:
In cooperative phase, use any sequence that would yield payoff vector v
Any deviation triggers uncooperative phase - NE of stage game with payoff vector w
If v>w, the uncooperative phase is the threat that sustains v, although it is not NE outcome
Minimum effort is an example of
strategic complementarity (the higher the action of the other player, the higher is my best response)
One NE risk dominates another NE in a 2x2 game if
the expected payoff of playing the risk dominant equilibrium strategy against a belief (½, ½ = full uncertainty) is higher than the expected payoff of the alternative equilibrium strategy.
Bayes rule
how to update beliefs rationally after receiving a signal (partial info) of a type; when there are multiple types and each type has possibly different probability of sending the signal.
Hence Buyes rule is: what is the probability that the signal is a support for a certain type (hypothesis) instead of an alternative hypothesis.
Pr(H|E) = Pr(E|H) Pr(H) / Pr(E|H) Pr(H)+Pr(E|not H) Pr(not H)
Types of incomplete information games
Adverse selection
Moral hazard
Adverse selection
type of the informed player affects the payoffs of the uninformed player; the uninformed player doesn’t observe directly the type, only the action taken by the type. It takes place when the BR of the informed players drives out some types out of the interaction and this impact is negative for economic efficiency.
Moral Hazard
the action of the informed player affects the payoffs of the uninformed player and the uninformed player does not directly observe this action, only some imprecise signal of the action.
Signaling
the informed player takes a costly action, more info is revealed if the costs of the action differ across types, so that only certain types send certain signals.
Signaling can work if the player types who hold private info can be distinguished by their strategy. The signal has to be costly, with costs differing across types.
Solutions to the adverse selection
Signaling
Screening
Screening
uninformed player designs a set of options giving incentives to the informed player to reveal own type (e.g. deductibles in insurance)
Types of Bayesian NE:
Pooling and separating
Pooling NE
if all types of the informed player choose the same action in equilibrium, implying that the uninformed player cannot distinguish the player types of the other players by observing the action chosen in the equilibrium (Aa, Bb)
Separating NE
if each type of the informed player chooses another action in equilibrium, implying that the uninformed player is able to distinguish the player types of the other player by observing the action chosen in the equilibrium. (Ab, Ba)
