1. Dynamic Games of Complete Information.

2. 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. Credibility
3. Commitment
4. The idea of Backward Induction

3. 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?

4. 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?

5. 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

6. Extensive form trees Rules for forming trees 1. Single starting point
2. No cycles
3. One way to proceed
Define precedence relation: a b a precedes b
1. ‘ ‘ is asymmetric: a b, means b a
2. ‘ ’ is transitive
3. x/ x and x// x implies x/ x// or x// x/
4. There is single initial node

7. 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

8. 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

9. 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

10. 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

11. 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 it
1. 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

12. 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”)

13. 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

14. 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)

15. 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

16. 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, δt2(1-x))
This is an infinite-horizon game of perfect info

17. 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

18. Subgame perfect equilibrium
When I makes offer, 2 will accept if he gets more than δ 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≤ This implies that, ,
Again, by symmetry,

19. 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.

20. 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 steps
3. It costs $2/7/15 to move 1/2/3 steps
4. 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

21. 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

22. 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

23. 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 

24. 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?

25. 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!!

26. 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)

27. 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

28. 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

29. 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 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

30. 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),

31. 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)

32. 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≥

33. 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
t 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+