Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Game Theory: Dynamic

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank Cowell Frank Cowell

November 2006 November 2006

Almost essential Game Theory: Strategy and Equilibrium

Prerequisites

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Overview...

Game and subgame

Equilibrium Issues

Applications

Game Theory: Dynamic

Mapping the temporal structure of games

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Time Why introduce “time” into model of a game?Why introduce “time” into model of a game?

““Time” involves structuring economic decisions Time” involves structuring economic decisions model the sequence of decision makingmodel the sequence of decision making consideration of rôle of information in that sequenceconsideration of rôle of information in that sequence

Be careful to distinguish Be careful to distinguish strategiesstrategies and and actionsactions see this in a simple settingsee this in a simple setting

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

A simple gameStage 1: Alf’s decisionStage 2: Bill’s decision following [LEFT]

The payoffs

(1a,1

b)

Alf

Bill Bill

[LEFT] [RIGHT]

[left] [right] [left] [right]

Stage 2: Bill’s decision following [RIGHT]

(2a,2

b) (3a,3

b) (4a,4

b)

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

[left-right][left-right]

(4a,4

b)

(1a,1

b)

[left-left][left-left]

(3a,3

b)

(1a,1

b)

(4a,4

b)(3a,3

b)

[RIG

HT]

[RIG

HT]

(2a,2

b)(2a,2

b)Alf

Alf

BillBill[right-left][right-left] [right-right][right-right]

A simple game: Normal formAlf has two strategiesBill has four strategiesThe payoffs

[LEFT][LEFT]

Always play [right] whatever Alf chooses

Always play [left] whatever Alf chooses

Play [left] if Alf plays [LEFT]. Play [right] if

Alf plays [RIGHT]

Play [right] if Alf plays [LEFT]. Play [left] if Alf plays [RIGHT]

Alf moves first: strategy set contains just two elements Bill moves second: strategy set contains four elements

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

The setting

Begin with some remindersBegin with some reminders

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Structure: 1 (reminders) Decision nodes Decision nodes

in the extensive form…in the extensive form… ……represent points where a decision is made by a player represent points where a decision is made by a player

Information setInformation set where is player (decision maker) located?where is player (decision maker) located? may not have observed some previous move in the may not have observed some previous move in the

gamegame knows that he is at one of a number of decision nodesknows that he is at one of a number of decision nodes collection of these nodes is the information setcollection of these nodes is the information set

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Structure: 2 (detail) StageStage

a particular point in the logical time sequence of the gamea particular point in the logical time sequence of the game payoffs come after the last stagepayoffs come after the last stage

Direct successor nodesDirect successor nodes take the decision branches (actions) that follow from node * take the decision branches (actions) that follow from node * if the branches lead to other decision nodes at the next stage…if the branches lead to other decision nodes at the next stage… ……then these are then these are direct successor nodes direct successor nodes to node *to node *

Indirect successor nodesIndirect successor nodes repeat the above through at least one more stage…repeat the above through at least one more stage… … … to get to get indirect successor nodesindirect successor nodes

How can we use this structure?How can we use this structure? break overall game down… break overall game down… ……into component games?into component games?

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Subgames (1) A A subgamesubgame of an extensive form game of an extensive form game

a subset of the game…a subset of the game… ……satisfying three conditionssatisfying three conditions

1.1. Begins with a “singleton” information setBegins with a “singleton” information set contains a single decision nodecontains a single decision node just like start of overall gamejust like start of overall game

2.2. Contains all the decision nodes that… Contains all the decision nodes that… are direct or indirect successors,…are direct or indirect successors,… ……and no othersand no others

3.3. If a decision node is in the subgame then If a decision node is in the subgame then any other node in the same information set…any other node in the same information set… ……is also in the subgame.is also in the subgame.

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Alf

[LEFT][LEFT] [RIGHT]

Alf

[LEFT] [RIGHT]

123456

78

Alf

[LEFT] [RIGHT] [RIGHT]

Alf

Alf

[LEFT] [RIGHT]

[left] [right] [left] [right]

Bill Bill

Subgames (2)Stage 1: (Alf’)Stage 2: (Bill)

The payoffsAdd a Stage (Alf again)...

A subgame...

Another subgame...

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Alf

[LEFT][LEFT] [RIGHT]

Alf

[LEFT] [RIGHT]

Alf

[LEFT] [RIGHT] [RIGHT]

Alf

Alf

[LEFT] [RIGHT]

[left] [right] [left] [right]

Bill Bill

Subgames (3)The previous structureAdditional strategy for AlfAmbiguity at stage 3A subgameNot a subgame (violates 2)[MID]

Alf

[LEFT] [RIGHT]

[left] [right] [left] [right]

Alf

[LEFT][LEFT] [RIGHT]

Alf

[LEFT]

Alf

[LEFT] [RIGHT]

Bill Bill

[RIGHT]

[left] [right]

Bill

Alf

123456789101112

Alf

Not a subgame (violates 3)

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Game and subgame: lessons ““Time” imposes structure on decision-makingTime” imposes structure on decision-making Representation of multistage gamesRepresentation of multistage games

requires carerequires care distinguish actions and strategiesdistinguish actions and strategies normal-form representation can be awkwardnormal-form representation can be awkward

Identifying subgamesIdentifying subgames three criteriathree criteria cases with non-singleton information sets are trickycases with non-singleton information sets are tricky

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Overview...

Game and subgame

Equilibrium Issues

Applications

Game Theory: Dynamic

Concepts and method

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Equilibrium Equilibrium raises issues of concept and methodEquilibrium raises issues of concept and method

both need some care…both need some care… ……as with the simple single-shot gamesas with the simple single-shot games

ConceptConcept can we use the Nash Equilibrium again?can we use the Nash Equilibrium again? clearly requires careful specification of the strategy setsclearly requires careful specification of the strategy sets

MethodMethod a simple search technique?a simple search technique? but will this always work?but will this always work?

We start with an outline of the method…We start with an outline of the method…

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Backwards induction Suppose the game has Suppose the game has NN stages stages Start with stage Start with stage NN

suppose there are suppose there are mm decision nodes at stage decision nodes at stage NN Pick an arbitrary node Pick an arbitrary node ……

suppose suppose hh is player at this stage is player at this stage ……determine determine hh’s choice, conditional on arriving at that node’s choice, conditional on arriving at that node ……note payoff to note payoff to hh and to every other player arising from this choice and to every other player arising from this choice

Repeat for each of the other Repeat for each of the other mm−1−1 nodes nodes this completely solves stage this completely solves stage NN gives gives mm vectors of [ vectors of [11],…,[],…,[mm]]

Re-use the values from solving stage Re-use the values from solving stage NN gives the payoffs for a game of gives the payoffs for a game of NN−1−1 stages stages

Continue on up the tree…Continue on up the tree…An example:

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Backwards induction: example Examine the last stage of the 3-stage game used earlierExamine the last stage of the 3-stage game used earlier Suppose the eight payoff-levels for Alf satisfy Suppose the eight payoff-levels for Alf satisfy

υυ11aa > > υυ22

aa (first node) (first node) υυ33

aa > > υυ44aa (second node) (second node)

υυ55aa > > υυ66

aa (third node) (third node) υυ77

aa > > υυ88a a (fourth node)(fourth node)

If the game had in fact reached the first node:If the game had in fact reached the first node: obviously Alf would choose [LEFT]obviously Alf would choose [LEFT] so value to (Alf, Bill) of reaching first node is [so value to (Alf, Bill) of reaching first node is [υυ11] = (] = (υυ11

aa,,υυ11bb))

Likewise the value of reaching other nodes at stage 3 is…Likewise the value of reaching other nodes at stage 3 is… [[υυ33] (second node)] (second node) [[υυ55] (third node)] (third node) [[υυ77] (fourth node)] (fourth node)

Backwards induction has reduced the 3-stage game…Backwards induction has reduced the 3-stage game… ……to a two-stage game to a two-stage game ……with payoffs [with payoffs [υυ11], [], [υυ33], [], [υυ55], [], [υυ77]]

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Alf

[LEFT][LEFT] [RIGHT]

Alf

[LEFT] [RIGHT]

Alf

[LEFT] [RIGHT] [RIGHT]

Alf

Alf

[LEFT] [RIGHT]

[left] [right] [left] [right]

Bill Bill

Backwards induction: diagram3-stage game as before

Payoffs to 3-stage game

Alf would play [LEFT] at this node

υ1a > υ2

a υ3

a > υ4a

υ5a > υ6

a υ7

a > υ8a

1

2

3

4 5 6 7

8

1

…and here

3

5 7

The 2-stage game derived from the 3-stage game

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Equilibrium: questions Backwards induction is a powerful methodBackwards induction is a powerful method

accords with intuitionaccords with intuition usually leads to a solutionusually leads to a solution

But what is the appropriate underlying concept?But what is the appropriate underlying concept? Does it find all the relevant equilibria?Does it find all the relevant equilibria? What is the role for the Nash Equilibrium (NE) What is the role for the Nash Equilibrium (NE)

concept?concept?

Begin with the last of these…Begin with the last of these… A simple example:

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Equilibrium example

(0,0) (2,1) (1,2) (1,2)

[LEFT] [RIGHT]

[left] [right] [left] [right]

The extensive formBill’s choices in final stageValues found through backwards inductionAlf’s choice in first stage

Backwards induction finds equilibrium payoff of 2 for Alf, 1 for BillBill(2,1) (1,2)

The equilibrium pathAlf

Bill

(2,1)

But what is/are the NE here?

Look at the game in normal form…

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

[left-right][left-right]

1,2

0,0

[left-left][left-left]

1,2

0,0

1,21,2

[RIG

HT]

[RIG

HT]

2,12,1

ss22 aa

Alf

Alf

BillBill

[right-left][right-left] [right-right][right-right]

Equilibrium example: Normal formAlf’s two strategiesBill’s four strategiesPayoffsss44

bbss11bb ss22

bb ss33bb

[LEFT][LEFT]

ss11 aa

Best replies to s1a

Best reply to s3b or to s4

b

Nash equilibria: (s1a,s3

b), (s1a,s4

b)

Best replies to s2a

Best reply to s1b or to s2

b

(s2a,s1

b), (s2a,s2

b),

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Equilibrium example: the set of NE

The set of NE include the solution already foundThe set of NE include the solution already found backwards induction method… (s1

a, s3b) yields payoff (2,1)

(s1a, s4

b) yields payoff (2,1) What of the What of the otherother NE? NE?

(s2a, s1

b) yields payoff (1,2) (s2

a, s2b) yields payoff (1,2)

These suggest two remarks First, Bill’s equilibrium strategy may induce some odd behaviour Second could such an NE be sustained in practice?

We follow up each of these in turn

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Equilibrium example: odd behaviour?

(0,0) (2,1) (1,2) (1,2)

[LEFT] [RIGHT]

[left] [right] [left] [right]

Bill

Alf

Bill

Take the Bill strategy Take the Bill strategy ss11b b = [left-left]= [left-left]

““Play [left] whatever Alf does”Play [left] whatever Alf does” If Alf plays [RIGHT] on MondayIf Alf plays [RIGHT] on Monday

On Tuesday it’s sensible for Bill to play On Tuesday it’s sensible for Bill to play [left][left]

But if Alf plays [LEFT] on MondayBut if Alf plays [LEFT] on Monday what should Bill do on Tuesday?what should Bill do on Tuesday? the above strategy says play [left]the above strategy says play [left] but, from Tuesday’s perspective, it’s oddbut, from Tuesday’s perspective, it’s odd

Given that the game reaches node *Given that the game reaches node * Bill then does better playing [right]Bill then does better playing [right]

Yet… Yet… ss11bb is part of a NE?? is part of a NE??

Tuesday

Monday

*

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Equilibrium example: practicality Again consider the NE not found by backwards inductionAgain consider the NE not found by backwards induction

Give a payoff of 1 to Alf, 2 to BillGive a payoff of 1 to Alf, 2 to Bill Could Bill “force” such a NE by a threat?Could Bill “force” such a NE by a threat?

Imagine the following pre-play conversation.Imagine the following pre-play conversation. Bill: “I will play strategy [left-left] whatever you do”Bill: “I will play strategy [left-left] whatever you do” Alf: “Which means?”Alf: “Which means?” Bill: “To avoid getting a payoff of 0 you had better play [RIGHT]”Bill: “To avoid getting a payoff of 0 you had better play [RIGHT]”

The weakness of this is obviousThe weakness of this is obvious Suppose Alf goes ahead and plays [LEFT]Suppose Alf goes ahead and plays [LEFT] Would Bill now really carry out this threat?Would Bill now really carry out this threat? After all Bill would also suffer (gets 0 instead of 1)After all Bill would also suffer (gets 0 instead of 1)

Bill’s threat seems incredibleBill’s threat seems incredible So the “equilibrium” that seems to rely on it is not very impressiveSo the “equilibrium” that seems to rely on it is not very impressive

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Equilibrium concept Some NEs are odd in the dynamic contextSome NEs are odd in the dynamic context

So there’s a need to refine equilibrium conceptSo there’s a need to refine equilibrium concept

Introduce Introduce Subgame-Perfect Nash Equilibrium Subgame-Perfect Nash Equilibrium (SPNE) (SPNE)

A profile of strategies is a SPNE for a game if itA profile of strategies is a SPNE for a game if it ……is a NEis a NE ……induces actions consistent with NE in every subgameinduces actions consistent with NE in every subgame

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

NE and SPNE All SPNE are NEAll SPNE are NE

reverse is not truereverse is not true some NE that are not SPNE involve agents making threats that are some NE that are not SPNE involve agents making threats that are

not crediblenot credible Definition of SPNE is demandingDefinition of SPNE is demanding

it says something about it says something about allall the subgames the subgames Even if some subgames are unlikely to be actually reached in Even if some subgames are unlikely to be actually reached in

practice practice Backward induction method is usefulBackward induction method is useful

but not suitable for all games with richer information setsbut not suitable for all games with richer information sets

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Equilibrium issues: summary Backwards induction provides a practical methodBackwards induction provides a practical method Also reveals weakness of NE concept Also reveals weakness of NE concept Some NE may imply use of empty threatsSome NE may imply use of empty threats

given a node where a move by given a node where a move by hh may damage opponent may damage opponent but would cause serious damage but would cause serious damage hh himself himself hh would not rationally make the move would not rationally make the move threatening this move should the node be reached is unimpressivethreatening this move should the node be reached is unimpressive

Discard these as candidates for equilibria?Discard these as candidates for equilibria? Focus just on those that satisfy subgame perfection Focus just on those that satisfy subgame perfection

See how these work with two applicationsSee how these work with two applications

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Overview...

Game and subgame

Equilibrium Issues

Applications

Game Theory: Dynamic

Industrial organisation (1)

•Market leadership•Market entry

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Market leadership: an output game Firm 1 (leader) gets to move first: chooses Firm 1 (leader) gets to move first: chooses qq¹ ¹ Firm 2 (follower) observes Firm 2 (follower) observes qq¹ and then chooses ¹ and then chooses qq² ² Nash Equilibria?Nash Equilibria?

given firms’ reaction function given firms’ reaction function any (any (qq¹, ¹, qq²) satisfying ²) satisfying qq² = ² = χχ²(²(qq¹) and ¹) and qq¹= ¹= χχ11((qq22) is the outcome of ) is the outcome of

a NEa NE many such NE involve incredible threatsmany such NE involve incredible threats

Find SPNE by backwards inductionFind SPNE by backwards induction start with follower’s choice of start with follower’s choice of qq² as best response to ² as best response to qq¹¹ this determines reaction function this determines reaction function χχ²(²(∙∙) ) given given χχ² the leader's profits are ² the leader's profits are pp((qq² + ² + χχ²(²(qq¹))¹))qq¹ ¹ − − CC¹(¹(qq¹)¹) max this w.r.t. max this w.r.t. qq¹ to give the solution¹ to give the solution

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Market leadership

q1

q2

2(·)

0

(qS, qS)

1 2

Leader max profits using follower’s stage-2 response

Follower’s isoprofit curvesFollower max profits conditional on q1

Leader’s opportunity setLeader’s isoprofit curves

Firm 2’s reaction function gives set of NE

Stackelberg solution as SPNE

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Overview...

Game and subgame

Equilibrium Issues

Applications

Game Theory: Dynamic

Industrial organisation (2)

•Market leadership•Market entry

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Entry: reusing an example

Take the example used to illustrate equilibriumTake the example used to illustrate equilibrium recall the issue of non-credible threatsrecall the issue of non-credible threats

Modify this for a model of market entryModify this for a model of market entry Suppose a monopolist faces possibility of another firm entering Suppose a monopolist faces possibility of another firm entering

will there be a fight (e.g. a price war) after entry?will there be a fight (e.g. a price war) after entry? should such a fight be threatened?should such a fight be threatened?

Replace Alf with Replace Alf with the potential entrant firmthe potential entrant firm [LEFT] becomes “enter the industry”[LEFT] becomes “enter the industry” [RIGHT] becomes “stay out”[RIGHT] becomes “stay out”

Replace Bill with Replace Bill with the incumbent firmthe incumbent firm [left] becomes “fight a potential entrant”[left] becomes “fight a potential entrant” [right] becomes “concede to a potential entrant”[right] becomes “concede to a potential entrant”

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Entry: reusing an example (more)

Payoffs: potential entrant firmPayoffs: potential entrant firm if it enters and there’s a fight: if it enters and there’s a fight: 00 if stays out: if stays out: 11 (profit in some alternative opportunity) (profit in some alternative opportunity) if enters and there’s no fight: if enters and there’s no fight: 22

Payoffs: incumbent firmPayoffs: incumbent firm if it fights an entrant: if it fights an entrant: 00 if concedes entry without a fight: if concedes entry without a fight: 11 if potential entrant stays out: if potential entrant stays out: 22 (monopoly profit) (monopoly profit)

Use the equilibrium developed earlierUse the equilibrium developed earlier Find the SPNEFind the SPNE We might guess that outcome depends on “strength” of the two firmsWe might guess that outcome depends on “strength” of the two firms Let’s see…Let’s see…

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

(0,0) (2,1) (1,2) (1,2)

[LEFT] [RIGHT]

[left] [right] [left] [right]

Bill

Alf

Bill

Entry example

(0,0) (2,1) (1,2) (1,2)

[IN][OUT]

[fight] [concede] [fight] [concede]

The modified version

Incumbent’s choice in final stage

Remove part of final stage that makes no sense

Entrant’s choice in first stage

SPNE is clearly (IN, concede)

A threat of fighting would be incredible

Inc

(1,2)

The equilibrium path

Ent

Inc

(2,1)

The original example

(2,1)

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Entry: modifying the example The simple result of the SPNE in this case is strikingThe simple result of the SPNE in this case is striking

but it rests on an assumption about the “strength” of the incumbentbut it rests on an assumption about the “strength” of the incumbent suppose payoffs to the incumbent in different outcomes are altered…suppose payoffs to the incumbent in different outcomes are altered… specifically, suppose that it’s relatively less costly to fightspecifically, suppose that it’s relatively less costly to fight what then?what then?

Payoffs of the potential entrant just as beforePayoffs of the potential entrant just as before if it enters and there’s a fight: if it enters and there’s a fight: 00 if stays out: if stays out: 11 if enters and there’s no fight: if enters and there’s no fight: 22

Lowest two payoffs for incumbent are interchangedLowest two payoffs for incumbent are interchanged if it fights an entrant: if it fights an entrant: 1 1 (maybe has an advantage on “home ground”)(maybe has an advantage on “home ground”) if concedes entry without a fight: if concedes entry without a fight: 0 0 (maybe dangerous to let newcomer (maybe dangerous to let newcomer

establish a foothold)establish a foothold) if potential entrant stays out: if potential entrant stays out: 22 (monopoly profit) (monopoly profit)

Take another look at the game and equilibrium…Take another look at the game and equilibrium…

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

(0,1) (2,0)

[IN] [OUT]

[fight] [concede](1,2)

Inc(1,2)(0,1)

Ent

Entry example (revised)

Incumbent’s choice in final stage

Entrant’s choice in first stage

The equilibrium path is trivial

SPNE involves potential entrant choosing [OUT]

The example revised

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Entry model: development Approach has been inflexibleApproach has been inflexible

relative strength of the firms are just hardwired into the payoffsrelative strength of the firms are just hardwired into the payoffs can we get more economic insight?can we get more economic insight?

What if the rules of the game were amended? What if the rules of the game were amended? could an incumbent make credible threats? could an incumbent make credible threats?

Introduce a “commitment device” Introduce a “commitment device” example of this is where a firm incurs sunk costsexample of this is where a firm incurs sunk costs the firm spends on an investment with no resale valuethe firm spends on an investment with no resale value

A simple version of the commitment idea A simple version of the commitment idea introduce an extra stage at beginning of the gameintroduce an extra stage at beginning of the game incumbent might carry out investment that costs incumbent might carry out investment that costs kk advertising?advertising?

First, generalise the

example:

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Entry deterrence: two subgames

[FIGHT] [CONCEDE]

1

[In] [Out]

2

(F ,0) (J, J)

(M, )_

Payoffs if there had been pre-play investment

Firm 2 chooses whether to enter Firm 1 chooses whether to fight

ΠM : monopoly profit for incumbent

Π > 0: reservation profit for challenger

ΠJ : profit for each if they split the market

ΠF : incumbent’s profit if there’s a fight

(J – k, J)

(M – k, )_

Investment cost k hits incumbent’s profits at each stage

Now fit the two subgames together

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Entry deterrence: full model

[FIGHT`] [CONCEDE]

1 1

1

[NOT INVEST]

[In] [Out] [In] [Out]

2

[INVEST]

[FIGHT`] [CONCEDE]

(F ,0) (J, J) (J– k, J)

(M, )_ (M – k, )_

(F ,0)

Firm 1 chooses whether to invest Firm 2 chooses whether to enter

2

Firm 1 chooses whether to fight

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Entry deterrence: equilibrium Suppose the Suppose the incumbent has committed to investment: incumbent has committed to investment:

if challenger enters… if challenger enters… it’s more profitable for incumbent to fight than concede ifit’s more profitable for incumbent to fight than concede if ΠΠF F > > ΠΠJ J – – kk

Should the Should the incumbent precommit to investment?incumbent precommit to investment? it pays to do this rather than just allow the no-investment subgame if …it pays to do this rather than just allow the no-investment subgame if … ……profit from deterrence exceeds that available without investment:profit from deterrence exceeds that available without investment: ΠΠMM – – kk > > ΠΠJJ

The SPNE is (INVEST, out) if:The SPNE is (INVEST, out) if: both both ΠΠF F > > ΠΠJ J – – kk and and ΠΠMM – – kk > > ΠΠJJ. . i.e. if i.e. if kk satisfies satisfies ΠΠJJ – – ΠΠF F < < kk < < ΠΠMM – – ΠΠJ J in this case deterrence “works”in this case deterrence “works”

So it So it may be may be impossible for the incumbent to deter entryimpossible for the incumbent to deter entry in this case if 2in this case if 2ΠΠJJ > > ΠΠMM + + ΠΠF …F … ……then there is no then there is no kk that will work that will work

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Summary New conceptsNew concepts

SubgameSubgame Subgame-perfect Nash EquilibriumSubgame-perfect Nash Equilibrium Backwards inductionBackwards induction Threats and credibilityThreats and credibility Commitment Commitment

What next?What next? Extension of time idea…Extension of time idea… ……repeated gamesrepeated games