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. Credibility 3. Commitment 4. The idea of Backward Induction. The Stackelberg game.
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Dynamic Games of Complete Information
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1. Information structure: who knows what and when?
2. Credibility
3. Commitment
4. The idea of Backward Induction
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
1. Single starting point
2. No cycles
3. One way to proceed
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
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 withvalues 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 Setsh 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
si(hi) єA(hi) for all hi єHi
A behavior strategy for i, denoted bi, is an element of Cartesian product
In a game of perfect recall, mixed and behavior strategies are equivalent
A finite game of perfect information has a pure strategy Nash equilibrium
1. Starts at a single node x of T
2. Contains all successors of x
3. If x/єG, and x//єh(x/), then x//єG
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”)
where
ui: Hk+1→R
ui(si , s-i)≥ ui(s/i , s-i) for all s/i
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
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
As δ1→1, for fixed δ2, we have v1→1, and player 1 gets entire pie.
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
- 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
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
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) Thenand
c) Firm 1’s expected profit at is
d)Sincefors≤t, impliesso 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
= (θV-c2)/(r+ θ),and for a 2nd stage duopolist is
W2 =(θV-c2)/(r+ 2θ),
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≥
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 . Straightforwardextension 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+