MICROECONOMICS Principles and Analysis Frank Cowell July 2015 1 Almost essential Game Theory Dynamic Prerequisites Note the detail in slides marked can only be seen if you run the slideshow ID: 386991
Download Presentation The PPT/PDF document "Repeated Games" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Repeated Games
MICROECONOMICSPrinciples and Analysis Frank Cowell
April 2018
1
Almost essential Game Theory: Dynamic
PrerequisitesSlide2
Overview
Basic structure
Equilibrium issues
Applications
Repeated Games
Embedding the game in context
April 2018
2Slide3
Introduction
Another examination of the role of timeDynamic analysis can be difficultmore than a few stagescan lead to complicated analysis of equilibrium
We need an alternative approachone that preserves basic insights of dynamic games
for example, subgame-perfect equilibriumBuild on the idea of dynamic games
introduce a jump move from the case of comparatively few stagesto the case of arbitrarily manyApril 2018
3Slide4
Repeated games
The alternative approachtake a series of the same gameembed it within a time-line structureBasic idea is simple
connect multiple instances of an atemporal game
model a repeated encounter between the players in the same situation of economic conflict Raises important questionshow does this structure differ from an
atemporal model?how does the repetition of a game differ from a single play?how does it differ from a collection of unrelated games of identical structure with identical players?April 2018
4Slide5
History
Why is the time-line different from a collection of unrelated games?The key is historyconsider history at any point on the timeline contains information about actual play
information accumulated up to that pointHistory can affect the nature of the game
at any stage all players can know all the accumulated informationstrategies can be conditioned on this informationHistory can play a role in the equilibrium
some interesting outcomes aren’t equilibria in a single encounterthese may be equilibrium outcomes in the repeated gamethe game’s history is used to support such outcomesApril 2018
5Slide6
Repeated games: Structure
The stage game take an instant in time
specify a simultaneous-move gamepayoffs completely specified by actions within the game
Repeat the stage game indefinitelythere’s an instance of the stage game at time 0,1,2,…,
t,… the possible payoffs are also repeated for each tpayoffs at t depends on actions in stage game at
t
A modified strategic environment
all previous actions assumed as common knowledge
so
agents’ strategies can be conditioned on this information
Modifies equilibrium behaviour and outcome?
April 2018
6Slide7
Equilibrium
Simplified structure has potential advantageswhether significant depends on nature of stage gameconcern nature of equilibriumPossibilities for equilibrium new strategy combinations supportable as equilibria?long-term cooperative outcomes
absent from a myopic analysis of a simple gameRefinements of subgame
perfection simplify the analysis:can rule out empty threats and incredible promisesdisregard irrelevant “might-have-beens
”April 20187Slide8
Overview
Basic structure
Equilibrium issues
Applications
Repeated Games
Developing the basic concepts
April 2018
8Slide9
Equilibrium: an approach
Focus on key question in repeated games:how can rational players use the information from history?need to address this to characterise equilibriumIllustrate a method in an argument by exampleoutline for the Prisoner's Dilemma gamesame players face same outcomes from their actions that they may choose in periods 1, 2, …,
t, … Prisoner's Dilemma particularly instructive given: its importance in microeconomics
pessimistic outcome of an isolated round of the game
April 20189Slide10
[RIGHT]
1,
1
3,
0
0,3
2,2
[LEFT]
Alf
Bill
[left]
[right]
Prisoner’s dilemma: Reminder
Payoffs in stage game
If Alf plays
[RIGHT]
Bill’s best response is
[right]
If Bill plays
[right]
Alf’s best response is
[
RIGHT
]
Nash Equilibrium
Outcome that Pareto dominates NE
The highlighted NE is inefficient
Could the Pareto-efficient outcome be an equilibrium in the repeated game?
Look at the structure
April 2018
10
* detail on slide can only be seen if you run the slideshowSlide11
Repeated Prisoner's dilemma
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
[left]
[right]
(2,2)
(0,3)
(3,
0
)
(1,
1
)
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
[left]
[right]
(2,2)
(0,3)
(3,
0
)
(1,
1
)
2
1
Stage game between (t=1)
Stage game (t=2) follows here
or here
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
[left]
[right]
(2,2)
(0,3)
(3,
0
)
(1,
1
)
2
or here
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
[left]
[right]
(2,2)
(0,3)
(3,
0
)
(1,
1
)
2
or here
Repeat this structure indefinitely…?
April 2018
11
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
[left]
[right]
(2,2)
(0,3)
(3,
0
)
(1,
1
)
2
* detail on slide can only be seen if you run the slideshowSlide12
Repeated Prisoner's dilemma
… … …
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
[left]
[right]
(2,2)
(0,3)
(3,
0
)
(1,
1
)
… … …
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
[left]
[right]
(2,2)
(0,3)
(3,
0
)
(1,
1
)
t
1
The stage game
repeated though time
Let's look at the detail
April 2018
12Slide13
Repeated PD: payoffs
To represent possibilities in long run:first consider payoffs available in the stage gamethen those available through mixturesIn the one-shot game payoffs simply representedit was enough to denote them as 0,…,3
purely ordinalarbitrary monotonic changes of the payoffs have no effect
Now we need a generalised notationcardinal values of utility matterwe need to sum utilities, compare utility differences
Evaluation of a payoff stream:suppose payoff to agent h in period t is uh(t
)
value of (
u
h
(1),
uh
(2),…, uh(t)…) is given by ∞
[1d] ∑ dt
1uh(t) t
=1where d is a discount factor 0 < d < 1April 2018
13Slide14
PD: stage game
A generalised notation for the stage gameconsider actions and payoffsin each of four fundamental casesBoth socially irresponsible: they play [RIGHT], [right]
get (
ua,
ub) where ua >
0,
u
b
>
0
Both socially responsible: they play [LEFT],[left] get (u*a, u
*b) where u*a > ua, u
*b > ub Only Alf socially responsible: they play [LEFT], [right]
get ( 0,`ub) where `
ub > u*b Only Bill socially responsible:
they play [RIGHT], [left] get (`ua, 0)
where `ua > u*a
A diagrammatic view
April 2018
14Slide15
Repeated Prisoner’s dilemma payoffs
*
u
a
u
b
0
(
u
a
,
u
b
)
•
(
u
*a
, u
*b
)
•
u
b
_
u
a
_
Space of utility payoffs
Payoffs for Prisoner's Dilemma
•
•
Nash-Equilibrium payoffs
Payoffs available through mixing
Feasible, superior points
"Efficient" outcomes
Payoffs Pareto-superior to NE
April 2018
15Slide16
Choosing a strategy: setting
Long-run advantage in the Pareto-efficient outcomepayoffs (u*a,
u*b) in each period
clearly better than (
ua, ub) in each periodSuppose the agents recognise the advantage
what actions would guarantee them this?
clearly they need to play [LEFT], [left] every period
The problem is lack of trust:
they cannot trust each other
nor indeed themselves:
Alf tempted to be antisocial and get
payoff`
ua by playing [RIGHT]Bill has a similar temptation April 2018
16Slide17
Choosing a strategy: formulation
Will a dominated outcome still be inevitable?Suppose each player adopts a strategy that rewards the other party's responsible behaviour by responding with the action [left] punishes antisocial behaviour with the action [right], thus generating the minimax payoffs
(
ua,
ub) Known as a trigger strategy Why the strategy is powerfulpunishment applies to
every
period after the one where the antisocial action occurred
if punishment invoked offender is “minimaxed for ever”
Look at it in detail
April 2018
17Slide18
[RIGHT]
Anything else
Bill’s action in 0,…,
t
Alf’s action at
t
+1
Repeated PD: trigger strategies
Take situation at t
First type of history
Response of other player to continue this history
Second type of history
Punishment response
[LEFT]
[left][left],…,[left]
[right]
Anything else
Alf’s action in 0,…,
t
Bill’s action at
t
+1
[left]
[LEFT][LEFT],…,[LEFT]
Will it work?
s
T
a
s
T
b
Trigger strategies
[
s
T
a
,
s
T
b
]
April 2018
18Slide19
Will the trigger strategy “work”?
Utility gain from “misbehaving” at t: `ua
−
u*a
What is value at t of punishment from t + 1 onwards?Difference in utility per period: u*
a
−
u
a
Discounted value of this in period t + 1: V := [
u*a − u
a]/[1 −d
]Value of this in period t: dV
= d[u*a
− ua]/[1
−d ]So agent chooses not to misbehave if `
ua − u*
a ≤ d[u*a
− ua ]/[1 −d ]But this is only going to work for specific parameters
value of
d
relative to
`
u
a
,
u
a
and
u
*
a
What values of discount factor will allow an equilibrium?
April 2018
19Slide20
Discounting and equilibrium
For an equilibrium condition must be satisfied for both a and bConsider the situation of a
Rearranging the condition from the previous slide:
d[
u*a − u
a
] ≥ [1
−
d]
[
`ua −
u*a ]
d[`ua
− ua ] ≥ [`
ua − u*a
] Simplifying this the condition must be
d ≥ da
where da := [`
ua − u*a
] / [`ua − ua ]
A similar result must also apply to agent b
Therefore we must have the condition:
d
≥
d
where
d
:= max {
d
a
,
d
b
}
April 2018
20Slide21
Repeated PD: SPNE
Assuming d ≥ d
, take the strategies [sTa
, sTb] prescribed by the Table
If there were antisocial behaviour at t consider subgame that would start at t + 1Alf could not increase his payoff by switching from [RIGHT] to [LEFT], given that Bill is playing [left] a similar remark applies to Bill
so strategies imply a NE for this
subgame
likewise for any subgame starting after
t
+ 1
But if [LEFT],[left] has been played in every period up till t:Alf would not wish to switch to [RIGHT]a similar remark applies to Bill
again we have a NESo, if d is large enough, [sTa,
sTb] is a Subgame-Perfect Equilibriumyields the payoffs (u
*a, u*b) in every periodApril 2018
21Slide22
Folk Theorem
The outcome of the repeated PD is instructiveillustrates an important resultthe Folk TheoremStrictly speaking a class of resultsfinite/infinite gamesdifferent types of equilibrium conceptsA standard version of the Theorem:
for a two-person infinitely repeated game:suppose discount factor is sufficiently close to 1
any combination of actions observed in any finite number of stages this is the outcome of a subgame-perfect equilibrium
April 201822Slide23
Assessment
The Folk Theorem central to repeated gamesperhaps better described as Folk Theorems a class of resultsClearly has considerable attractionPut its significance in contextmakes relatively modest claims gives a possibility result
Only seen one example of the Folk Theoremlet’s apply itto well known oligopoly examples
April 2018
23Slide24
Overview
Basic structure
Equilibrium issues
Applications
Repeated Games
Some well-known examples
April 2018
24Slide25
Cournot competition: repeated
Start by reinterpreting PD as Cournot duopolytwo identical firmsfirms can each choose one of two levels of output – [high] or [low]can firms sustain a low-output (i.e. high-profit) equilibrium?
Possible actions and outcomes in the stage game:
[HIGH], [high]: both firms get Cournot-Nash payoff PC
> 0[LOW], [low]: both firms get joint-profit maximising payoff PJ > P
C
[HIGH], [low]: payoffs are (
`P,
0) where
`P >
PJ Folk theorem: get SPNE with payoffs (PJ
, PJ) if d is large enoughCritical value for the discount factor d is
`P − PJ
d = ──────
`P − PCBut we should say more
Let’s review the standard Cournot diagramApril 2018
25Slide26
q
1
q
2
c
2
(
·
)
c
1
(
·
)
0
(
q
C
,
q
C
)
1 2
(
q
J
,
q
J
)
1 2
Cournot stage game
Firm 2’s Iso-profit curves
Firm 2’s reaction function
Cournot-Nash equilibrium
Firm 1’s
Iso
-profit curves
Firm 1’s reaction function
Outputs with higher profits for both firms
Joint profit-maximising solution
Output that forces other firm’s profit to 0
`
q
1
`
q
2
April 2018
26Slide27
Repeated Cournot game: Punishment
Standard Cournot model is richer than simple PD:action space for PD stage game just has the two output levels continuum of output levels introduces further possibilitiesMinimax profit level for firm 1 in a
Cournot duopolyis
zero, not the NE outcome P
Carises where firm 2 sets output to `q2 such that 1 makes no profit Imagine a deviation by firm 1 at time t
raises
q
1
above joint profit-max level
Would minimax be used as punishment from
t + 1 to ∞?
clearly (0,`q2) is not on firm 2's reaction functionso cannot be best response by firm 2 to an action by firm 1
so it cannot belong to the NE of the subgameeverlasting minimax punishment is not credible in this caseApril 2018
27Slide28
Repeated Cournot game: Payoffs
P
1
P
2
0
(P
C
,
P
C
)
P
•
•
P
Space of profits for the two firms
Cournot-Nash outcome
Joint-profit maximisation
(P
J
,P
J
)
Minimax outcomes
Payoffs available in repeated game
Now review Bertrand competition
April 2018
28Slide29
p
2
c
c
p
M
p
M
p
1
Bertrand stage game
Firm 1’s reaction function
Firm 2’s reaction function
Marginal cost pricing
Monopoly pricing
Nash equilibrium
April 2018
29Slide30
Bertrand competition: repeated
NE of the stage game:set price equal to marginal cost c
results in zero profits NE outcome is the minimax outcome
minimax outcome is implementable as a Nash equilibrium in all the subgames following a defection from cooperation
In repeated Bertrand competitionfirms set pM if acting “cooperatively”split profits between themif one firm deviates from thisothers then set price to
c
Repeated Bertrand: result
can enforce joint profit
maximisation
through trigger strategyprovided discount factor is large enough
April 2018
30Slide31
Repeated Bertrand game: Payoffs
P
1
P
2
0
P
M
•
•
P
M
Space of profits for the two firms
Bertrand-Nash outcome
Firm 1 as a monopoly
Firm 2 as a monopoly
Payoffs available in repeated game
April 2018
31Slide32
Repeated games: summary
New concepts:Stage gameHistoryThe Folk TheoremTrigger strategyWhat next?Games under uncertainty
April 2018
32