1. MICROECONOMICS Principles and Analysis Frank Cowell
Prerequisites
Almost essential
Game Theory: Dynamic
Repeated Games
MICROECONOMICS
Principles and Analysis
Frank Cowell
Note: the detail in slides marked “ * ” can only be seen if you run the slideshow
July 2015
Frank Cowell: Moral Hazard

2. Overview Embedding the game in context Repeated Games Basic structure
Equilibrium issues
Applications
July 2015

3. Introduction Another examination of the role of time
Dynamic analysis can be difficult
more than a few stages
can lead to complicated analysis of equilibrium
We need an alternative approach
one that preserves basic insights of dynamic games
for example, subgame-perfect equilibrium
Build on the idea of dynamic games
introduce a jump
move from the case of comparatively few stages
to the case of arbitrarily many
July 2015

4. Repeated games The alternative approach Basic idea is simple
take a series of the same game
embed it within a time-line structure
Basic 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 questions
how 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?
July 2015

5. History
Why is the time-line different from a collection of unrelated games?
The key is history
consider history at any point on the timeline
contains information about actual play
information accumulated up to that point
History can affect the nature of the game
at any stage all players can know all the accumulated information
strategies can be conditioned on this information
History can play a role in the equilibrium
some interesting outcomes aren’t equilibria in a single encounter
these may be equilibrium outcomes in the repeated game
the game’s history is used to support such outcomes
July 2015

6. Repeated games: Structure
The stage game
take an instant in time
specify a simultaneous-move game
payoffs completely specified by actions within the game
Repeat the stage game indefinitely
there’s an instance of the stage game at time 0,1,2,…,t,…
the possible payoffs are also repeated for each t
payoffs 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?
July 2015

7. Equilibrium Simplified structure has potential advantages
whether significant depends on nature of stage game
concern nature of equilibrium
Possibilities for equilibrium
new strategy combinations supportable as equilibria?
long-term cooperative outcomes
absent from a myopic analysis of a simple game
Refinements of subgame perfection simplify the analysis:
can rule out empty threats
and incredible promises
disregard irrelevant “might-have-beens”
July 2015

8. Overview Developing the basic concepts Repeated Games Basic structure
Equilibrium issues
Applications
July 2015

9. 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 equilibrium
Illustrate a method in an argument by example
Outline for the Prisoner's Dilemma game
same 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
July 2015

10. *Prisoner’s dilemma: Reminder
Payoffs in stage game
If Alf plays [RIGHT] then Bill’s best response is [right]
Alf
[LEFT]
2,2
0,3
If Bill plays [right] then Alf’s best response is [RIGHT]
Nash Equilibrium
[RIGHT]
3,0
1,1
Outcome that Pareto dominates NE
[left]
[right]
The highlighted NE is inefficient
Could the Pareto-efficient outcome be an equilibrium in the repeated game?
Look at the structure
Bill
July 2015

11. *Repeated Prisoner's dilemma
Stage game between (t=1)
Stage game (t=2) follows here 1
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
(2,2)
(0,3)
(3,0)
(1,1)
or here
or here
or here
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
(2,2)
(0,3)
(3,0)
(1,1) 2
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
(2,2)
(0,3)
(3,0)
(1,1) 2
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
(2,2)
(0,3)
(3,0)
(1,1) 2
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
(2,2)
(0,3)
(3,0)
(1,1) 2
Repeat this structure indefinitely…?
July 2015

12. Repeated Prisoner's dilemma
The stage game 1
… … …
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
(2,2)
(0,3)
(3,0)
(1,1)
repeated though time
… … …
Bill
Alf
[LEFT]
[RIGHT]
[left]
[right]
(2,2)
(0,3)
(3,0)
(1,1) t
Let's look at the detail
July 2015

13. Repeated PD: payoffs To represent possibilities in long run:
first consider payoffs available in the stage game
then those available through mixtures
In the one-shot game payoffs simply represented
it was enough to denote them as 0,…,3
purely ordinal
arbitrary monotonic changes of the payoffs have no effect
Now we need a generalised notation
cardinal values of utility matter
we 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 (uh(1), uh(2),…, uh(t)…) is given by ∞
[1d] ∑ dt1uh(t)
t=1
where d is a discount factor 0 < d < 1
July 2015

14. PD: stage game A generalised notation for the stage game
consider actions and payoffs
in each of four fundamental cases
Both socially irresponsible:
they play [RIGHT], [right]
get ( ua, ub) where ua > 0, ub > 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
July 2015

15. Repeated Prisoner’s dilemma payoffs
Space of utility payoffs
Payoffs for Prisoner's Dilemma ua ub
Nash-Equilibrium payoffs
Payoffs Pareto-superior to NE
Payoffs available through mixing ub _
Feasible, superior points •
"Efficient" outcomes •
( u*a, u*b ) 𝕌* •
( ua, ub ) • ua _
July 2015

16. Choosing a strategy: setting
Long-run advantage in the Pareto-efficient outcome
payoffs (u*a, u*b) in each period
clearly better than ( ua, ub) in each period
Suppose 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
July 2015

17. 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 powerful
punishment 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
July 2015

18. Repeated PD: trigger strategies
sTa
Bill’s action in 0,…,t
Alf’s action at t+1
Take situation at t
First type of history
[left][left],…,[left]
[LEFT]
Response of other player to continue this history
Anything else
Second type of history
[RIGHT]
Punishment response
Trigger strategies [sTa, sTb]
sTb
Alf’s action in 0,…,t
Bill’s action at t+1
Will it work?
[left]
[LEFT][LEFT],…,[LEFT]
Anything else
[right]
July 2015

19. 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 − ua
Discounted value of this in period t + 1: V := [u*a − ua]/[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 `ua, ua and u*a
What values of discount factor will allow an equilibrium?
July 2015

20. Discounting and equilibrium
For an equilibrium condition must be satisfied for both a and b
Consider the situation of a
Rearranging the condition from the previous slide:
d[u*a − ua ] ≥ [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 {da , db}
July 2015

21. Repeated PD: SPNE
Assuming d ≥ d, take the strategies [sTa, sTb] prescribed by the Table
If there were antisocial behaviour at t consider the subgame that would then start at t + 1
Alf 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]
again we have a NE
So, if d is large enough, [sTa, sTb] is a Subgame-Perfect Equilibrium
yields the payoffs (u*a, u*b) in every period
July 2015

22. Folk Theorem The outcome of the repeated PD is instructive
illustrates an important result
the Folk Theorem
Strictly speaking a class of results
finite/infinite games
different types of equilibrium concepts
A 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
July 2015

23. Assessment The Folk Theorem central to repeated games
perhaps better described as Folk Theorems
a class of results
Clearly has considerable attraction
Put its significance in context
makes relatively modest claims
gives a possibility result
Only seen one example of the Folk Theorem
let’s apply it
to well known oligopoly examples
July 2015

24. Overview Some well-known examples Repeated Games Basic structure
Equilibrium issues
Applications
July 2015

25. Cournot competition: repeated
Start by reinterpreting PD as Cournot duopoly
two identical firms
firms 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 > PC
[HIGH], [low]: payoffs are (`P, 0) where `P > PJ
Folk theorem: get SPNE with payoffs (PJ, PJ) if d is large enough
Critical value for the discount factor d is
`P − PJ
d = ──────
`P − PC
But we should say more
Let’s review the standard Cournot diagram
July 2015

26. Cournot stage game q2 `q2 c1(·) c2(·) q1 `q1 1 2 (qC, qC) 1 2 (qJ, qJ)
Firm 1’s Iso-profit curves q2
Firm 2’s Iso-profit curves
Firm 1’s reaction function
Firm 2’s reaction function
Cournot-Nash equilibrium
Outputs with higher profits for both firms
`q2
c1(·)
Joint profit-maximising solution
Output that forces other firm’s profit to 0
(qC, qC)
c2(·)
(qJ, qJ) q1
`q1
July 2015

27. 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 possibilities
Minimax profit level for firm 1 in a Cournot duopoly
is zero, not the NE outcome PC
arises where firm 2 sets output to `q2 such that 1 makes no profit
Imagine a deviation by firm 1 at time t
raises q1 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 function
so cannot be best response by firm 2 to an action by firm 1
so it cannot belong to the NE of the subgame
everlasting minimax punishment is not credible in this case
July 2015

28. Repeated Cournot game: Payoffs
Space of profits for the two firms
Cournot-Nash outcome P2
Joint-profit maximisation
Minimax outcomes  P •
Payoffs available in repeated game
(PJ,PJ)
(PC,PC)
Now review Bertrand competition P1 •  P
July 2015

29. Bertrand stage game p2 pM c p1 c pM Marginal cost pricing
Monopoly pricing
Firm 1’s reaction function
Firm 2’s reaction function
Nash equilibrium pM c p1 c pM
July 2015

30. 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 competition
firms set pM if acting “cooperatively”
split profits between them
if one firm deviates from this
others then set price to c
Repeated Bertrand: result
can enforce joint profit maximisation through trigger strategy
provided discount factor is large enough
July 2015

31. Repeated Bertrand game: Payoffs
Space of profits for the two firms
Bertrand-Nash outcome P2
Firm 1 as a monopoly
Firm 2 as a monopoly PM •
Payoffs available in repeated game P1 • PM
July 2015

32. Repeated games: summary
New concepts:
Stage game
History
The Folk Theorem
Trigger strategy
What next?
Games under uncertainty
July 2015