MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Game Theory Strategy and Equilibrium Prerequisites July 2015 1 Introduction Presentation builds on Game Theory Strategy and Equilibrium ID: 550014
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Slide1
Games: Mixed Strategies
MICROECONOMICSPrinciples and Analysis Frank Cowell
Almost essential Game Theory: Strategy and Equilibrium
Prerequisites
April 2018
1Slide2
Introduction
Presentation builds on Game Theory: Strategy and EquilibriumPurpose is to: extend the concept of strategy extend the characterisation of the equilibrium of a gamePoint of taking these steps: tidy up loose ends from elementary discussion of equilibrium
lay basis for more sophisticated use of gamessome important applications in economicsApril 20182Slide3
Overview
The problem
Mixed strategies
Applications
Games: Equilibrium
An introduction to the issues
April 2018
3Slide4
Games: a brief review
Components of a gameplayers (agents) h = 1,2,…objectives of players rules of play
outcomesStrategysh: a complete plan for all positions the game may reachSh: the set of all possible sh
focus on “best response” of each playerEquilibriumelementary but limited concept – dominant-strategy equilibriummore general – Nash equilibriumNE each player is making the best reply to everyone else
April 2018
4Slide5
NE: An important result
In some cases an important result applieswhere strategy sets are infinite for example where agents choose a value from an interval
THEOREM: If the game is such that, for all agents h, the strategy sets Sh are convex, compact subsets of
and the payoff functions vh are continuous and quasiconcave, then the game has a Nash equilibrium in pure strategies
Result is similar to existence result for General Equilibrium
April 2018
5Slide6
A problem?
Where strategy sets are finiteagain we may wish to seek a Nash Equilibriumbased on the idea of best replyBut some games apparently have no NE
example – the discoordination gameDoes this mean that we have to abandon the NE concept?Can the solution concept be extended? how to generaliseto encompass this type of problemFirst, a brief review of the exampleApril 2018
6Slide7
[–]
2,1
0,3
1,2
3,0
[+]
Player
a
Player
b
[+]
[–]
“Discoordination”
If a plays
[–]
then b’s best response is
[+]
If b plays
[+]
then a’s best response is
[
+
]
If a plays
[+]
then b’s best response is
[
–
]
Apparently, no Nash equilibrium!
Again there’s more to the Nash-equilibrium story here
(to be continued)
If b plays
[
–
]
then a’s best response is
[
–
]
Discoordination
This game may seem no more than a frustrating chase round the payoff table. The two players’ interests are always opposed (unlike Chicken or the Battle of the Sexes). But it is an elementary representation of class of important economic models. An example is the tax-audit game where Player 1 is the tax authority (“audit”, “no-audit”) and Player 2 is the potentially cheating taxpayer (“cheat”, “no-cheat”). More on this later.
Story
April 2018
7
* detail on slide can only be seen if you run the slideshowSlide8
Overview
The problem
Mixed strategies
Applications
Games: Equilibrium
Extending
the strategy concept
April 2018
8Slide9
A way forward
Extend the concept of strategynew terminology required Pure strategythe type of strategy that has been discussed so far
a deterministic plan for every possible eventuality in the gameMixed strategya probabilistic approach to playderived from set of pure strategiespure strategies themselves can be seen as special cases of mixed strategiesApril 2018
9Slide10
Mixed strategies
For each player take a set of pure strategies SAssign to each member of S a probability p that it will be playedEnables a “
convexification” of the problemThis means that new candidates for equilibrium can be foundand some nice results can be establishedbut we need to interpret this with careApril 201810Slide11
Strategy space
– extended?Use the example of strategy space in Game Theory: Basics In the simplest case
S is just two blobs “Left” and “Right”
L
Suppose we introduce the probability
p
Could it effectively change the strategy space like this?
R
S
This is misleading
There is no “half-left” or “three-quarters-right” strategy
Try a different graphical representation
April 2018
11Slide12
Strategy – a representation
Draw a diagram in the space of the probabilitiesStart by enumerating each strategy in the set Sif there are n
of these we’ll need an n-dimensional diagramdimension i corresponds to the probability that strategy i is playedThen plot the points (1,0,0,…), (0,1,0,…),
(0,0,1,…),… Each point represents the case where the corresponding pure strategy is playedTreat these points like “radio buttons”: you can only push one down at a timelikewise the points (1,0,0,…), (0,1,0,…),
(0,0,1,…),… are mutually exclusiveLook at this in the case
n
= 2
April 2018
12Slide13
Two pure strategies in
S
p
R
p
L
(1,0)
(0,1)
Cases where
0 <
p
< 1
Probability of playing
R
Playing L with certainty
Probability of playing
L
p
L
+
p
R
= 1
Pure strategy means being at one of the two points (1,0) or (0,1)
Playing R with certainty
But what of the other points?
April 2018
13Slide14
Mixed strategy – a representation
The endpoints (1,0) and (0,1) represent playing the “pure” strategies L and R
Any point on the line joining them represents a probabilistic mixture of L and R:middle of the line represents case where the person spins a fair coin before choosing L or R
pL = pR = ½Consider the extension to the case of 3 pure strategies:
strategies consist of the actions “Left”, “
M
iddle”, “
R
ight”
we now have three “buttons”
(1,0,0)
,
(0,1,0)
,
(0,0,1)
Again consider the diagram:April 2018
14Slide15
Three pure strategies in
S
0
p
R
p
L
p
M
Third axis corresponds to probability of playing “
M
iddle”
(
1,0,0
)
(0,0,1)
(0,1,0)
Three “buttons” for the three pure strategies
Introduce possibility of having
0 <
p
< 1
p
L
+
p
M
+
p
R
= 1
April 2018
15Slide16
Strategy space again
Allowing for the possibility of “mixing”Player’s strategy space consists of a pair:a collection of pure strategies (as before)a collection of probabilitiesOf course this applies to each of the players in the gameHow does this fit into the structure of the game?
Two main issues:modelling of payoffsmodelling and interpretation of probabilitiesApril 201816Slide17
The payoffs
We need to take more care herea question of the nature of “utility”If pure strategies only are relevantpayoffs can usually be modelled simply
usually can be represented in terms of ordinal utilityIf players are acting probabilisticallyconsider how to model prospective payoffstake into account preferences under uncertaintyuse expected utility?Cardinal versus ordinal utility
if we take expectations over many cells of the payoff table……we need a cardinal utility conceptcan transform payoffs u only by scale and origin: a + bu
otherwise expectations operator is meaningless
April 2018
17Slide18
Probability and payoffs
Expected utility approach induces a simple structureWe can express resulting payoff assum of
(utility associated with each button) (probability each button is pressed)So we have a neat linear relationshippayoff is linear in utility associated with each buttonpayoff is linear in probabilitiestherefore payoff is linear in strategic variables
Implications of this structure?April 201818Slide19
Reaction correspondence
A simple tool build on the idea of the reaction function used in oligopolygiven competitor’s quantity, choose your own quantityBut, because of linearity need a more general concept
reaction correspondencemultivalued at some pointsallows for a “bang-bang” solutionGood analogies with simple price-taking optimisationthink of demand-response with straight-line indifference curvesor straight-line isoquantsComputation of equilibrium need not be difficult
April 201819Slide20
Mixed strategies: computation
To find optimal mixed-strategy:take beliefs about probabilities used by other playerscalculate expected payoff as function of these and one’s own probabilities
find response of expected payoff to one’s own probabilitycompute reaction correspondenceTo compute mixed-strategy equilibriumtake each agent’s reaction correspondence
find equilibrium from intersection of reaction correspondencesPoints to notebeliefs about others’ probabilities are crucialstage 4 above often leads to p = 0 or p = 1
acts like a kind of tipping mechanism
April 2018
20Slide21
Mixed strategies: result
The linearity of the problem permits us to close a gap We have another existence result for Nash EquilibriumTHEOREM Every game with a finite number of pure strategies has an equilibrium in mixed strategies
April 201821Slide22
The random variable
Key to the equilibrium concept: probabilityBut what is the nature of this entity?an explicit generating model?subjective idiosyncratic probability?will others observe and believe the probability?How is one agent’s probability related to another?
do each choose independent probabilities?or is it worth considering a correlated random variable?Examine these issues using two illustrationsApril 201822Slide23
Overview
The problem
Mixed strategies
Applications
Games: Equilibrium
An example where only a mixed strategy can work
The audit game
Chicken
April 2018
23Slide24
Illustration: the audit game
Builds on the idea of a discoordination gameA taxpayer chooses whether or not to report income ypays tax ty if reports
pays 0 if does not report and concealment is not discoveredpays tax plus fine F if does not report and concealment is discoveredTax authority (TA) chooses whether or not to audit taxpayerincurs resource cost c if it auditsreceives due tax
ty plus fine F if concealment is discoveredExamine equilibriumfirst demonstrate no equilibrium in pure strategiesthen the mixed-strategy equilibriumFirst examine best responses of each player to the other
April 2018
24Slide25
[report]
[1
t
]
y
,
ty
[1
t
]
y
,
ty
c
y
, 0
[1
t]
y F,
ty + F c
[conceal]
Taxpayer
Tax Authority
[Audit]
[Not audit]
Audit game: normal form
Each chooses one of two actions
(taxpayer, TA) payoffs
If taxpayer conceals then TA will audit
If taxpayer reports then TA won’t audit
[
1
t
]
y
>
[1
t
]
y
F
t
y
+
F
c
> 0
If TA audits then taxpayer will report
t
y
c
>
t
y
y >
[1
t
]
y
If TA doesn’t audit, taxpayer will conceal
No equilibrium in pure strategies
mixed strategies
April 2018
25Slide26
Audit game: mixed strategy approach
Now suppose each player behaves probabilisticallytaxpayer conceals with probability pa TA audits with probability
pb Each player maximises expected payoffchooses own probabilitytaking as given the other’s probabilityFollow through this processfirst calculate expected payoffsthen compute optimal p given the other’s
pthen find equilibrium as a pair of probabilities April 201826Slide27
Audit game: taxpayer’s problem
Payoff to taxpayer, given TA’s value of pb
: if conceals: ua = pb [y
ty F] + [1 pb ] y =
y
p
b
ty
p
b
F
if reports:
u
a = y
ty If taxpayer selects a value of pa, calculate expected payoffE
ua = pa [y
pbty pb
F] + [1 pa ] [y
ty]= [1 t] y
+ pa [1 pb] ty
papbF Taxpayer’s problem: choose pa to max Eua Compute effect on
Eua of changing pa :
∂E
u
a
/ ∂
p
a
= [
1
p
b
]
ty
p
b
F
define
p
*b
=
ty
/ [
ty + F] then E u
a / ∂pa is positive if pb < p*b, negative if “>”So optimal strategy is set
pa to its max value 1 if pb is low (below p*b)set pa
to its min value 0 if pb is highApril 201827Slide28
Audit game: TA’s problem
Payoff to TA, given taxpayer’s value of pa:
if audits: ub = pa [ty + F
c] + [1 pa][ty
c] =
ty
c
+
p
a
F
if does not audit:
u
b
= pa
∙ 0 + [1 pa]
ty = [1 pa] ty
If TA selects a value of pb, calculate expected payoffE
ub = pb [ty
c + paF] + [1
pb] [1 pa] ty
= [1 pa ] ty + papb [
ty + F] pb
cTA’s problem: choose
p
b
to max
E
u
b
Compute effect on
E
u
b
of changing
p
b
:
∂
E
u
b
/ ∂
p
b
=
pa [ty
+ F] c define p*a = c / [
ty + F] then E ub / ∂pb is positive if
pa < p*a, negative if “>”So optimal strategy is set pb to its min value 0 if pa
is low (below p*a)set pb to its max value 1 if pa is high
April 201828Slide29
Audit game: equilibrium
1
0
1
p
b
p
a
p
*a
p
*b
•
The space of mixed strategies
Taxpayer’s reaction correspondence
•
•
p
a
= 1 if
p
b
<
p
*b
p
a
= 0 if
p
b
>
p
*b
TA’s reaction correspondence
p
b
= 0 if
p
a
<
p
*a
p
b
= 1 if
p
a
>
p
*a
Equilibrium at intersection
(
p
*
a
,
p
*b
)
April 2018
29Slide30
Overview
The problem
Mixed strategies
Applications
Games: Equilibrium
Mixed strategy or correlated strategy?
The audit game
Chicken
April 2018
30Slide31
Chicken game again
A number of possible background storiesthink of this as individuals’ contribution to a public projectthere’s the danger that one may contribute, while the other “free rides”
and the danger that nobody contributes at allbut this isn’t quite the classic “public good problem” (later)Two players with binary choices call them “contribute” and “not contribute”denote as [+] and [−]Payoff structureif you contribute and the other doesn’t, then you get 1 the other gets 3
if both of you contribute, then you both get 2if neither of you contribute, then you both get 0 First, let’s remind ourselves of pure strategy NE April 2018
31Slide32
[–]
0,0
3,1
1,3
2,2
[+]
Player
a
Player
b
[+]
[–]
Chicken game: normal form
If a plays
[–],
b’s best response is
[+]
If b plays
[+],
a’s best response is
[
–
]
Resulting NE
By symmetry, another NE
Two NE’s in pure strategies
Up to this point utility can be taken as purely ordinal
mixed strategies
April 2018
32Slide33
Chicken: mixed strategy approach
Each player behaves probabilistically:a plays [+] with probability pa
b plays [+] with probability pb Expected payoff to a is Eu
a = pa [2∙pb +1·[1
−pb]] + [1
−
p
a
][3
·
p
b
+ 0
·
[1
− pb]] =
pa +3pb − 2pa
pb Differentiating: dE
ua /dpa =1−
2pb which is positive (resp. negative) if pb <
½ (resp. pb > ½)So a’s optimal strategy is
pa =1 if pb < ½ ,
pa = 0 if pb > ½ Similar reasoning for bTherefore mixed-strategy equilibrium is
(pa,pb ) = (
½,½)
where payoffs are
(
u
a
,
u
b
) = (1
½
, 1
½)
April 2018
33Slide34
Chicken: payoffs
u
a
u
b
0
2
3
1
2
3
1
Space of utilities
Two NEs in pure strategies (as before)
Yellow area: possibilities if
utility is thrown away
Green area: utilities achievable by
randomisation
(1
½
, 1
½
):
Mixed-strategy NE
Red broken line: Efficient outcomes
Mid point: equitable and efficient solution?
Utility here must have
cardinal
significance
Equitable and efficient outcome found
by taking
½ each of the two pure-strategy NEs
How can we get this?
•
•
(1
½
, 1
½
)
•
•
April 2018
34Slide35
Chicken game: summary
If the agents move sequentially then get a pure-strategy NEoutcome will be either (3,1) or (1,3)depends on who moves first If move simultaneously: a coordination problem?
Randomisation by the two agents?independent action does not help muchproduces payoffs (1½, 1½)
But if they use the same randomisation device:play [+] with the same probability p expected payoff for each is u
a = p
+ 3
p
− [
2
p
]
2
= 4
p
[
1 − p ]
maximised where p = ½ Appropriate randomisation seems to solve the coordination problem April 2018
35Slide36
Another application?
Do mixed strategies this help solve Prisoner’s Dilemma? A reexaminationagain model as individuals’ contribution to a public project
two players with binary choices: contribute [+], not-contribute []close to standard public-good problemBut payoff structure crucially different from “chicken”if you contribute and the other doesn’t, you get 0 the other gets 3if both of you contribute, then you both get 2
if neither of you contribute, then you both get 1 We know the outcome in pure strategies:there’s a NE ([], [])but payoffs in NE are strictly dominated by those for ([+
], [+])Now consider mixed strategy
April 2018
36Slide37
PD: mixed-strategy approach
Again each player behaves probabilistically:a plays [+] with probability pa
b plays [+] with probability pb Expected payoff to a is Eua =
pa [2∙pb + 0·[1−p
b]] + [1−p
a
][3
·
p
b
+ 1
·
[1
−
p
b
]] = 1 + 2p
b − paclearly E
ua is decreasing in paOptimal strategies
from the above, a will set pa to its minimum value, 0
by symmetry, b will also set pb to 0So we are back to the non-cooperative solution :
(pa,pb ) = (
0,0)both play [] with certaintyMixed-strategy approach does not resolve the dilemma
April 201837Slide38
Assessment
Mixed strategy: a key development of game theorycloses a hole in the NE approachbut is it a theoretical artifice?Is mixed-strategy equilibrium an appropriate device? depends on the context of the microeconomic model
degree to which it’s plausible that agents observe and understand the use of randomisationNot the last word on equilibrium conceptsas extra depth added to the nature of gamenew refinements of definitionExample of further developmentsintroduction of time, in dynamic gamesintroduction of asymmetric information
April 201838