Dynamic gams with complete information III
Dynamic games with complete information III
ECOS3012 Lecture 6
September 14, 2021
Last week
Finitely repeated games
If a stage game G has a unique NE, then no matter how many times it is repeated, this NE is played in every stage game.
If a stage game G has multiple NE, then these NE can be used to generate credible reward for cooperation.
This week
Infinitely repeated games:
Even if the stage game has a unique NE, cooperation can be supported in SPE
A few questions:
Why infinitely repeated games?
How do we compute the payoffs?
What does a strategy look like?
How do we check whether a strategy profile is SPE?
One-shot deviation principle
How many SPE are there?
Folk theorem
Why infinitely repeated games?
Some of strategic interactions do not have a clear “final period”
Two firms competing
Two classmates decide whether to contribute to a group project: they may meet again
Two prisoners…
Infinitely repeated PD
What is the total payoff?
The sum of payoffs wouldn’t work any more: everything becomes infinity, and we cannot compare payoffs anymore
Prisoner 2
Confess Not Confess
Prisoner 1 Confess 1, 1 5, 0
Not Confess 0, 5 4, 4
The following prisoners’ dilemma game is repeated infinitely
Present value of payoff sequences
We discount future payoffs by a factor of .
Given the discount factor , the present value of the infinite sequence of payoffs is
Interpretations of the discount factor
Interest loss
Suppose interest rate per period is , then $1 tomorrow is equivalent to $ today
Caring the future less
Having to study for exams tomorrow does not feel as painful as having to do it today
Uncertainty of game continuation
After each stage is played, the game ends immediately with probability .
At time t, expected utility from the next stage is only
Math tip: geometric series
Calculate the value of a geometric series
Example:
What does a strategy look like?
A contingent plan
Specifies the action the player will take in each stage, for each possible history of play through the previous stage
Recall: when PD is repeated twice
What to do in Stage 1
What to do in Stage 2 if outcome of Stage 1 is (C, C)
What to do in Stage 2 if outcome of Stage 1 is (C, NC)
What to do in Stage 2 if outcome of Stage 1 is (NC, C)
What to do in Stage 2 if outcome of Stage 1 is (NC, NC)
Strategies in an infinitely repeated game
Yes, a lot of things to be specified
Therefore, a lot of strategies
Example: trigger strategy s
Play “Not confess” in the first stage
In Stage t,
If the outcomes of all t-1 preceding stages has been (NC, NC), then play “Not Confess
Otherwise, play “Confess”
Subgame perfect Nash equilibrium
Recall the definition:
A Nash equilibrium of a repeated game is subgame-perfect if players’ strategies constitute a Nash equilibrium in every subgame.
What is a subgame in a repeated game:
There is one subgame beginning at stage t+1 for each of the possible histories of play through stage t
A subgame is identified by its history
Four proper subgames in twice repeated PD game
How to check whether a strategy profile constitute an SPE
Backward Induction does not work any more: there is no final stage to start from.
Go back to the definition:
NE in every subgame
Problems:
Too many subgames: maybe we can categorize them
Too many possible deviation: one-shot deviation principle
One-shot deviation principle
Trigger strategy
Check whether (s, s) is SPE
Subgames where “confess” has been played before
Subgames where “confess” has not been played (including the game itself)
Prisoner 2
Confess Not Confess
Prisoner 1 Confess 1, 1 5, 0
Not Confess 0, 5 4, 4
Trigger strategy s
Play “Not confess” in the first stage
In Stage t,
If the outcomes of all t-1 preceding stages has been (NC, NC), then play “Not Confess
Otherwise, play “Confess”
Trigger strategy with 1-period punishment
Check whether (s’, s’) is SPE
Punishment phase:
Cooperation phase:
Prisoner 2
Confess Not Confess
Prisoner 1 Confess 1, 1 5, 0
Not Confess 0, 5 4, 4
Trigger strategy with 1-period punishment s’
Play “Not confess” in the first stage
In Stage t,
If some player deviated in the last period, play “Confess” and then restart the game by playing “Not Confess”.
Otherwise, play “Not Confess”
Comparison between the two SPE
Trigger strategy with infinite-period punishment: SPE if and only if .
Trigger strategy with one-period punishment: SPE if and only if .
For both strategy profiles, we need players to be patient enough.
Strategy profiles with heavier penalty: less requirement on players’ patience.
What payoffs can we possibly achieve?
(s, s) and (s’, s’) both give us (NC, NC) in every stage.
Payoff = 4 in every period.
Another SPE: Both players “Confess” regardless of the history of play.
(C, C) in every stage.
Payoff = 1 in every period.
But how many more?
Folk theorem: there are many SPE
Average payoffs
Recall that the present value of the infinite payoff sequence is
Suppose for all t, then the present value is .
We know that for this sequence, the average payoff is 4.
We multiply PV of all other sequences also by to get the average payoff
The average payoff of an infinite payoff sequence is
A value that is comparable to stage payoff
Feasible payoffs of a stage game
Minmax payoff: the lowest payoff one can get in a stage game
Idea: Suppose your opponent wants to punish you, how low can he push your payoff to be?
Take into consideration that you can also choose your own actions
You know that, your opponent also knows that
Prisoner 2
Confess Not Confess
Prisoner 1 Confess 1, 1 5, 0
Not Confess 0, 5 4, 4
Folk theorem
Folk Theorem: Infinitely repeated PD
Minmax payoff: another example
L R
U -2, 2 1, -2
M 1, -2 -2, 2
D 0, 1 0, 1
What is Player 1’s minmax payoff?
U -> 1 – 3q
M -> -2 + 3q
D -> 0
Minmax payoff = 0
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