Dynamic Games of Complete Information

.

Extensive form games

• To model games with a dynamic structure

• Main issues with a dynamic structure:1. Information structure: who knows what

and when?2. Credibility3. Commitment 4. The idea of Backward Induction

The Stackelberg game• A dynamic version of the Cournot game

• Player 1, “Stackelberg leader” chooses output q1 first

• Player 2, “Stackelberg follower” chooses output q2 next

• Demand is linear: p(q)=12-q

• Player i’s utility is ui(q1 , q2 )=[12- (q1 + q2)]qi

• What is the Stackelberg equilibrium?

• Are there other Nash equilibria?

Model of strategic investment

• Firms 1, 2 have average cost of 2 per unit

• Firm 1 can install new technology at cost f. Then average cost is zero

• Firm 2 can observe firm 1’s investment

• The two firms then move simultaneously to set quantity. Demand is p(q)=14-q

• How should firm 1 forecast its rival’s output?• Backward induction not directly applicable. Why?

The Extensive form

• Building blocks of the extensive form game:

1. The set of players

2. The order of moves - i.e. who moves when

3. The player’s payoffs as a function of moves

4. What the player’s choices are when they move

5. What a player knows when making his choice

6. Probability distribution over any exogenous events

Extensive form trees

• Rules for forming trees1. Single starting point2. No cycles3. One way to proceed

• Define precedence relation: a b a precedes b1. ‘ ‘ is asymmetric: a b, means b a2. ‘ ’ is transitive3. x/ x and x// x implies x/ x// or x// x/

4. There is single initial node

Formalizing the extensive form 1. Let i єI be the finite set of players

2. Let i(x) bet set of players that move at node x

3. Let Z be set of terminal nodes. Maps ui:Z→R with values ui(z) are i’s payoffs to a sequence of moves z

4. Let A(x) be set of feasible actions at node x

5. Information Sets h partition nodes of the tree:

a. Each node x is in only one information set h(x)

b. If x/єh(x), then player moving at x does not know if he is at x or x/

c. If x/єh(x), then the same player moves at x & x/

d. If x/єh(x), then A(x) = A(x/). Thus A(h) is action set at information set h

Example

• Two people want to go to a Broadway musical in great demand

• There is exactly one ticket left, and whoever arrives first gets it

• There are three transportation choices: c(cab); b(bus); s(subway)

• Player 1 leaves home a little earlier• A cab is faster than the subway, which is

faster than a bus

Strategies & equilibria in extensive form

• Let Hi be set of player i’s information sets

• Let be the set of all actions for i

• A pure strategy for i is a map si: Hi → Ai , with

si(hi) є A(hi) for all hi є Hi

• The set of pure strategies for i is Si=

• The number of i’s pure strategies is given by the product

• Mixed strategies in extensive form are called behavior strategies. Let ∆(A(hi)) be prob dist on A(hi)

A behavior strategy for i, denoted bi, is an element of Cartesian product

)( iHh

i hAAii

)( iHh

hAii

))((## iHh

i hASii

))(( iHh

hAii

Strategic-form versus extensive-form • Using its pure strategies and payoffs, an extensive form can be

transformed to strategic form• Extensive form interpretation: player i waits until hi is reached

before deciding how to play there• Strategic form interpretation: player i makes a complete

contingent plan in advance• Games of perfect information with all singleton information sets

constitute a special class• Any mixed strategy σi (strat form) generates a behavior strategy

bi (ext form), but many different σi’s can generate the same bi

• Theorem (Kuhn 1953):In a game of perfect recall, mixed and behavior strategies are equivalent

Backward induction & Subgame perfection• Theorem (Zermelo 1913; Kuhn 1953)

A finite game of perfect information has a pure strategy Nash equilibrium

• Subgame perfection is the analog of backward induction for multi-player situations

• G is a proper subgame of an extensive form game T if it1. Starts at a single node x of T

2. Contains all successors of x

3. If x/є G, and x//є h(x/), then x//є G

• A behavior strategy σ of an extensive form game is a subgame perfect equilibrium if the restriction of σ to G is a Nash equilibrium of G for every proper subgame G

Multi-stage games with observed actions

1. There are k stages: 0, 1, …, k-1

2. All players know the actions chosen at all previous stages

3. All players move simultaneously in each stage

4. This includes games where players move alternately (all other players have strategy: “do nothing”)

Multi-stage games with observed actions

• Let a0≡ be the stage-0 action-profile• At the beginning of stage1, players know history h1

which is just a0

• Let Ai(h1) be player i’s action set at stage 1 with history h1

• hk+1 is history at end of stage k, hk+1=(a0, a1,… ak), and Ai(hk+1) is player i’s action set at stage k+1

• If game is K stages, HK is set of all ‘terminal histories’• A pure strategy for i is seq. of maps such that

where

),...,,( 002

01 Iaaa

Kk

kis 0}{

)(: ki

kki HAHs )()( k

iHh

ki hAHA

kk

Multi-stage games with observed actions• Payoffs are defined on terminal histories,

ui: Hk+1→R

• In most applications, payoffs are additively separable over stages. This isn’t necessary

• The game from stage k on with history hk is a proper subgame G(hk), and a strategy profile s for whole game induces si│hk for subgame G(hk)

• A Nash equilibrium s satisfies the familiar condition ui(si , s-i)≥ ui(s/

i , s-i) for all s/i

• A Nash equilibrium s is subgame perfect if si│hk is a Nash equilibrium for every subgame G(hk)

Principle of optimality and subgame perfection

• For multi-stage games with observed actions, we have a useful characterization of subgame perfection for the finite-horizon case

• Theorem (One-stage-deviation principle)

A strategy profile ‘s’ is subgame perfect iff no player i can gain by deviating from ‘s’ in a single stage and conforming to ‘s’ thereafter

• This theorem can be extended to the infinite horizon case

Rubinstein Bargaining model

• Two players have to share a pie of size 1

• The game:

Step1: In periods 0, 2, 4,…, player 1 proposes a split (x, 1-x)

Step 2: If player 2 accepts in period 2k, game ends. If he rejects, he proposes (x, 1-x) in period 2k+1

Step 3: If player 1 accepts, game ends. Else, Step 1

• Discount factors are δ1, δ2, and if split (x, 1-x)

is accepted at time t payoffs are (δt1x, δt

2(1-x))

• This is an infinite-horizon game of perfect info

Subgame perfect equilibrium

• A continuation payoff of a strategy profile in subgame starting at t is utility in time-t units of outcome induced by that profile

• Let , be player 1’a lowest and highest continuation payoffs in any subgame that begins with player 1 making an offer

• Let , be player 1’a lowest and highest continuation payoffs in any subgame that begins with player 2 making an offer

• Similarly define for player 2

1v 1v

1w 1w

2222 , , , wwvv

Subgame perfect equilibrium

• When I makes offer, 2 will accept if he gets more than δ2 . Hence, . Also, by symmetry,

• Suppose player 1 makes offer (x, 1-x). If 2 accepts, the min he can get is , and therefore, 1-x≥ . Thus, x≤1- . This implies that, ,

• Again, by symmetry,

2v 221 1 vv

112 1 vv

22v 22v22v

221sup vx

221 1 vv

112 1 vv

Subgame perfect equilibrium• Now, , so,

• Similarly, , which gives,

• But, , and thus,

• Proceeding similarly,

• What is the effect of patience?

As δ1→1, for fixed δ2, we have v1→1, and player 1 gets entire pie.

)1(11 112221 vvv 21

21 1

1

v

)1(11 112221 vvv 21

21

1

1

v

11 vv 21

211 1

1

vv

21

122 1

1

vv

A model of R&D race• Firms R and S are conducting R&D• Several stages need to be completed• Simplifying assumptions:

1. Distance from goal can be measured. E.g. firm S is n-steps away from completion

2. Either firm can move 1, 2, or 3 steps3. It costs $2/7/15 to move 1/2/3 steps4. Firm completing all the steps first gets patent worth 20

• What would happen if R-S were a cartel, maximizing joint profits?- Since only one firm gets patent, only it does R&D- Chosen firm moves 1-step at a time, and firm closer to finishing is chosen

The extensive form of R&D game

• Suppose the firms take turns deciding on R&D investment: becomes a game of perfect info

• Converting to extensive-form:

1. Transform to location-space picture

2. Let (r, s) be coordinates of R and S , with r depicting how far R is from finishing

Subgame perfect equilibrium of R&D game

• If S is in R’s safety zone- whatever the zone number -it should drop out of the race

• Firm S in its own safety zone spends the minimum amount on R&D, moves one step at a time and wins the patent

• In Trigger zone n, each firm spends what it needs to- profitably -to get an advantage and move the game to its safety zone n-1

David vs Goliath in Entry Decisions

• Suppose Goliath has $700 and David has $300

• They are gambling types, and prefer roulette

• Whoever ends up with more money after the next round will win ultimately

• Suppose David moves first and makes the safest bet

• He can never win

David vs Goliath in Entry Decisions

• He should take one of the more risky gambles

• Bets $300 that the ball would land on a multiple of 3 – wins $900 w.p. 12/37

• What is Goliath’s best response?

• To exactly imitate David’s bet !

• Again, David can never win • Is there any hope for David?

David vs Goliath in Entry Decisions

• David should have gone second and differentiated himself

• This situation is parallel to new product launch decisions when a firm with shallow pockets competes against a firm with deep pockets

• If going second is not feasible, then entrant should take riskier bets – like launching a product with some chance of failing!!

Patent races• When is there competition or monopoly?• Depends on possibility of preemption and

leapfrogging • Not about chance of winning, but about chance of

being favorite• Consider two firms i=1,2, and let value of patent be V• Productivity of R&D increases over time• Let ωi(t) be firm i’s total experience at time t• μ(ωi(t)) is firm i’s hazard rate at t. Let μi(t)=μ(ωi(t)) • Discovery probability is an exponential waiting time• Note, discovery is stochastic and firm 2 (enters t2>0)

could discover before firm 1 (enters t1 =0)

Assumptions and preliminaries• Cost is c and common discount rate is r• Probability that no firm makes discovery before time t is

• Expected value of patent race for firm i is

• R&D is viable for monopoly,

• It is unprofitable for both firms to always do R&D

• State of competition is pair of experiences

]))()((exp[0 21 t

d

dtcVtdrt i

t

t

i

i

})(}{]))()(([exp{0 21

0})(}{])([exp{0

0

dtcVtdrtt

0})(}{])(2[exp{0

0

dtcVtdrtt

),(~21

Model without leapfrogging• Result 1 (є-preemption): In the unique subgame perfect

equilibrium, whatever t2, firm 1 engages in R&D and firm 2 drops out of the race.

• Sketch of proof:

i. Let Ω(ω) be firm 1’s experience such that it has zero payoff when firm 2 has experience ω, and both firms do R&D

ii. Show that a firm does R&D until discovery or drops out immediately

a) Suppose not. Let initial state be and both firms do R&D till time t and firm 1 drops out at t with zero profits

b) Then and

c) Firm 1’s expected profit at is

),(~21

)),(()(~22 ttt 0))(~()),(( 1221 ttt

~

dscVsdrst s

})(]}{)]()([[exp{)~( 12

0 0

11

]})]()([[exp{))(~( 2

0

11 drttt

Sketch of proof of Result 1 (contd.):

d) Since for s≤t, implies

so firm 1 would not join patent race: a contradiction

iii. The strategy: ‘If ωi≥ ωj, firm i stays in and j drops out iff

ωj ≤ Ω(ωi)’, is subgame perfect

iv. Suppose there exists t such that, conditional on neither having dropped out at t, firm 1 drops out with some prob.

v. Let be the supremum of such times for firm 1. for firm 2

vi. Claim:

vii. Claim: Firm 2 drops out with probability 1 at time

viii. Then, firm 1 will not drop out at time -є for small є

ix. Proceeding similarly, firm 1 never drops out

x. Then, firm 2 never does R&D

2t

1t

21 tt 1t

1t

0)(& ,0))(~( 11 cVst 0)~(1

Model with leapfrogging

• There are 2 stages: preliminary and final stages

• Costs are c1 and c2

• Time-t experience for firm i in stage j is ωji(t)

• Probability of making 1st, 2nd stage discovery at time t if it has not been made before are μi(t), θ

• Cannot accumulate 2nd stage experience without making 1st stage discovery

• Payoff for 2nd stage monopolist,

= (θV-c2)/(r+ θ), and for a 2nd stage duopolist is

W2 =(θV-c2)/(r+ 2θ),

dtcVeM tr )( 2

0

)(2

Model with leapfrogging

• Result 2: There exists a SPNE where the leading firm always does R&D unless the rival does R&D and completes the 1st stage before . The follower either, (i) drops out at the start, (ii) does R&D until , or (iii) always does R&D unless the leader passes the 1st stage before ( +t2)

1

1

1

Model with leapfrogging

• Sketch of proof of Result 2:

i. There are 2 decision points: one firm has finished 1st stage, or both are in 1st stage

ii. Let be level of 1st stage experience such that a firm with experience less (greater) than will drop out (stay in) even if the rival has completed stage 1. It is defined by

iii a. If 2 has completed stage 1, firm 1 should stay in if t ≥

b. If 1 has completed stage 1, firm 2 should stay in if t-t2≥

1

1

0})(}{)()(exp{ 121

00

1

dscWsdsrs

1

1

Model with leapfrogging• Sketch of proof of Result 2:

iv. Suppose neither has made 1st stage discovery at time

v. Firm 1 will hit experience level before firm 2. After that it will stay in.

vi. Can 2 do R&D profitably when both stay in forever? If yes, both do R&D unless firm 1 passes stage 1 before

t2+ . Else, the subgame from onwards resembles the no-leapfrogging situation. From result 1, 2 will drop at

vii. Consider race before time . Straightforward extension of arguments in Result 1 show that

- there is є-preemption and 2 quits at t=0

- firm 2 does R&D till t= t2+

1

1

1

1

1

1

1