ECOS3012 Lecture 5
Dynamic Games with Complete Information II
ECOS3012 Lecture 5
September 7, 2021
Last week
Dynamic games with one mover at each stage
Extensive form: game tree
Normal form: table
NE vs SPE (subgame perfect Nash equilibrium)
SPE is stronger than NE
Sometimes SPE can be too strong
Backward induction
Today
Multiple players move at the same stage
Repeated games
Finitely repeated games
Infinitely repeated games (next lecture)
Two other examples
Bank runs
Tariffs and international competition
Finitely repeated prisoner’s dilemma
Play this game twice
Both prisoners observe the outcome of the first play before the second play begin.
Payoff for the game: sum of the payoffs from the two stage games.
Prisoner 2
Confess Not Confess
Prisoner 1 Confess 1, 1 5, 0
Not Confess 0, 5 4, 4
Finitely repeated PD: normal form
How many pure strategies does each player have?
What do they look like?
How to represent PD in extensive form
Normal form to extensive form
Information set
When the play of the game reaches
a node in the information set, the
player does not know which node in
the information set has been reached
Subgames
Start from a singleton information set
Includes everything thereafter
Does not cut information set
Finitely repeated PD: extensive form
How many proper subgames are there?
Finitely repeated PD: SPE
Question: Since the prisoners will meet again in the future, can they somehow cooperate in the first period?
What is an SPE of this two-stage game?
Definition: NE in every subgame
(C, C) in every proper subgame
Finitely repeated PD: SPE
Move to Stage 1:
Regardless of the outcome of Stage 1, (C, C) will be played in Stage 2
(C, C) is the unique NE in the first stage
Prisoner 2
C in Stage 1 NC in Stage 1
Prisoner 1 C in Stage 1 1 +1, 1 +1 5 +1, 0 +1
NC in Stage 1 0 +1, 5 +1 4 +1, 4 +1
Finitely repeated PD: SPE
Unique SPE:
Unique SPE outcome: Both prisoners confess in both stages and get total payoff of (2, 2)
Takeaway messages
Having a future stage does not help.
“I’ll not confess if you don’t confess this time” – Not credible
Same conclusion even if prisoners play this game for a million times
Backward induction from the last stage
Both confess in the final stage; therefore both confess in the second-last stage, and therefor both confess in the third-last stage
Both confess in every stage
Generalizes to any finitely repeated game.
Problem: Unique NE for the stage game. No room for credible threats.
Finitely repeated game with multiple equilibria
Consider a different stage game:
Question: Can we construct a SPE in which prisoners play “not confess” in Stage 1?
Prisoner 2
Confess Not Confess Run
Prisoner 1 Confess 1, 1 5, 0 0, 0
Not Confess 0, 5 4, 4 0, 0
Run 0, 0 0, 0 3, 3
Finitely repeated game with multiple equilibria
How many subgames do we have?
In each subgame, a Nash equilibrium must be played. (does not have to be the same NE)
How many strategies does each player have?
A player’s strategy specifies what she does in the first stage, and what she does in the second stage given each outcome from stage 1.
Yes, given the player’s own strategy, some outcomes will never happen. But we still specify what to do in case they happened.
Finitely repeated game with multiple equilibria
Observation: the stage game has two pure NE: (Confess, Confess) and (Run, Run)
In any SPE, players must either play (C, C) or (R, R) in the second stage
Prisoner 2
Confess Not Confess Run
Prisoner 1 Confess 1, 1 5, 0 0, 0
Not Confess 0, 5 4, 4 0, 0
Run 0, 0 0, 0 3, 3
SPE to support (NC, NC) in first stage
Consider the following strategy (s):
“Not Confess” in stage 1
“Run” in stage 2 if stage 1 outcome is (NC, NC)
“Confess” in stage 2 if stage 1 out come is not (NC, NC)
How to best respond to this strategy:
In subgames where outcome is not (NC, NC): “Confess”
In subgames where outcome is (NC, NC): “Run”
Move to Stage 1:
C: 5 + 1
NC: 4 + 3
R: 0 + 1
SPE to support (NC, NC) in first stage
We have shown that both players playing strategy s is a subgame perfect Nash equilibrium
Outcome of this subgame perfect Nash equilibrium
(NC, NC) in stage 1 <- Not an NE of the stage game
(R, R) in stage 2
How did we construct strategy s?
Use the fact that there are two possible outcomes in stage 2
(C, C): the bad outcome
(R, R): the good outcome
Design strategies so that
If cooperate in stage 1, play the good outcome in stage 2
If not cooperate in stage 1, play the bad outcome in stage 2
Use the multiplicity of stage game equilibria to design reward/penalty, so that players have an incentive to cooperate.
Other examples
Bank run
Two investors: each deposited $100 with the bank
Investment takes two periods to mature
If investment matures: pays a total of $300 to the two investors
If investment liquidated before maturity: a total of $180 can be recoverd
Bank run
Stage 1
Stage 2
Investor 2
Withdraw Don’t
Investor 1 Withdraw 90, 90 100, 80
Don’t 80, 100 Next stage
Investor 2
Withdraw Don’t
Investor 1 Withdraw 150, 150 200, 100
Don’t 100, 200 150, 150
Bank run: backward induction to find SPE
Stage 2：
Unique NE: (withdraw, withdraw) with payoff (150, 150)
Stage 1:
Two pure-strategy subgame perfect outcomes:
Both withdraw in stage 1 and get payoff (90, 90) – “Bank run”
Both withdraw in stage 2 and get payoff (150, 150)
Lesson: This is a coordination game. Bank runs can arise in equilibrium even without any negative shock to the economy.
Investor 2
Withdraw Don’t
Investor 1 Withdraw 90, 90 100, 80
Don’t 80, 100 150, 150
Tariffs and international competition
Players: 2 governments, 2 firms
Timeline:
Stage 1: Two governments simultaneously set up tariff rates (t1, t2)
Stage 2: One firm in each country. Each firm produces good for both home consumption (h) and export (3). After observing (t1, t2), the firms simultaneously choose (h1, e1) and (h2, e2).
Payoffs:
Market-clearing price in country i: Pi (hi, ej) = 100 – hi – ej
Total production cost for firm i: Ci(hi, ei) = 10(hi + ei)
Firm I’s payoff is its profit
Tariffs and international competition
Tariffs and international competition
Tariffs and international competition
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