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## ECO290E: Game Theory

Lecture 9: Applications of Dynamic Games

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## Review of Lecture 9

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## Cournot Model: Revisit

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## Stackelberg Model

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## Stackelberg Model: Analysis (1) 

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62 DYNAMIC GAMES OF COMPLETE INFORMATION

q2 0; (3) the payoff to firm i is given by the profit function 7r;(q;, qj)

### =

q;[P(Q) - eJ,

where P(Q) = a - Q is the market-clearing price when the aggre- gate quantity on the m arket is Q

q,

### +

q2, and e is the constant marginal cost of production (fixed costs being zero).

To solve for the backwards-induction outcome of this game, we first compure firm 2's reaction to an arbitrary quantity by firm 1. R2(q,) solves

max 7r2(q" q2) = max q2[a - q, - q2 - eJ,

Q2?:O

which yields

a -

e R2(q,) = 2 '

provided

### q, <

a-e. The same equation for R2(q,) appeared in our analysiS of the simultaneous-move Cournot game in Sec- tion 1.2.A. The difference is tha t here R2(q,} is truly firm 2's reac- tion to firm l's observed quantity, whereas in the Cournot analYSis R2(q,) is firm 2's best response to a hypothesized quantity to be simultaneously chosen by firm 1.

Since firm 1 can solve firm 2's problem as well as firm 2 can solve it, firm 1 should anticipate that the quantity choice

### q,

will be met with the reaction Thus, firm l's problem in the first -S tage of the game amounts to

which yields

### qi

= -2-a-e and R2(qi) = -4-a-e

as the backwards-induction outcome of the Stackelberg duopoly game.'

4JUst as "Cournot equilibrium" and "Bertrand equilibrium" typically re- fer to the equillbria of the Coumot and Bertrand games, references to

Dynamic Games of Complete and Perfect Information

### 63

Recall from Chapter 1 that in the Nash equilibrium of the Cournot game each firm produces (a-e)/3. Thus, aggregate quan- tity in the backwards-induction outcome of the Stackelberg game, 3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game. In the Stackelberg gal):1e, how- ever, firm 1 could have chosen its Cournot quantity, (a - e)/3, in which case firm 2 would have responded with its Cournot quan- tity. Thus, in the Stackelberg game, firm 1 could have achieved its Cournot profit level but chose to do otherwise, so firm l's profit in the Stackelberg game must exceed its profit in the Cournot game. But the market-clearing price is lower in the Stackelberg game, so aggregate profits are lower, so the fact tha t firm 1 is better off im- plies that firm 2 is worse off in the Stackelberg than in the Cournot game.

The observation that firm 2 does worse in the Stackelberg than in the Cournot game illustrates an important difference between single- and multi-person decision problems. In single-person deci- sion theory, having more information can never make the decision maker worse off. In game .theory, however, having more informa- tion (or, more precisely, having it known to the other players that one has more information) ean make a player worse off.

In the Stackelberg game, the information in question is firm l 's quantity: firm 2 knows

### q"

and (as importantly) firm 1 knows that firm 2 knows

### q, .

To see the effect this information has, consider the modified sequential-move game in which firm 1 chooses

### q"

after which firm 2 chooses q2 but does so without observing

### q,.

If

firm 2 believes that firm 1 has chosen its Stackelberg quantity

### qi

= (a-e)/2, then firm 2's best response is again R2(qll = (a-e)/4. But if firm 1 anticipates that firm 2 will hold this belief a nd so choose this quaritity, then firm 1 prefers to choose its best response to (a - e)/4-namely, 3(a - e)/B-rather than its Stackelberg quantity (a-e)/2. Thus, firm 2 should not believe that firm 1 h as chosen its Stackelberg quantity. Rather, the unique Nash equilibrium of this

"Stackelberg equilibrium" often mean that the game is sequentia l- rather than simultaneous-move. As noted in the previous section, however, sequential-move games sometimes have multiple Nash equilibria, only one of which is associated with the backwards-induction outcome of the game. Thus, "Stackelberg equilib- rium" can refer both to the sequential-move nature of the game and to the use of a stronger solution concept than simply Nash equilibrium .

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## Stackelberg Model: Analysis (2) 

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62 DYNAMIC GAMES OF COMPLETE INFORMATION q2 0; (3) the payoff to firm i is given by the profit function

7r;(q;, qj)

### =

q;[P(Q) - eJ,

where P(Q) = a - Q is the market-clearing price when the aggre- gate quantity on the m arket is Q

q,

### +

q2, and e is the constant marginal cost of production (fixed costs being zero).

To solve for the backwards-induction outcome of this game, we first compure firm 2's reaction to an arbitrary quantity by firm 1. R2(q,) solves

max 7r2(q" q2) = max q2[a - q, - q2 - eJ,

Q2?:O

which yields

a -

e R2(q,) = 2 '

provided

### q, <

a-e. The same equation for R2(q,) appeared in our analysiS of the simultaneous-move Cournot game in Sec- tion 1.2.A. The difference is tha t here R2 (q,} is truly firm 2's reac- tion to firm l's observed quantity, whereas in the Cournot analYSis R2(q,) is firm 2's best response to a hypothesized quantity to be simultaneously chosen by firm 1.

Since firm 1 can solve firm 2's problem as well as firm 2 can solve it, firm 1 should anticipate that the quantity choice

### q,

will be met with the reaction Thus, firm l's problem in the first -S tage of the game amounts to

which yields

### qi

= a-e -2- and R2(qi) = a-e -4-

as the backwards-induction outcome of the Stackelberg duopoly game.'

4JUst as "Cournot equilibrium" and "Bertrand equilibrium" typically re- fer to the equillbria of the Coumot and Bertrand games, references to

Dynamic Games of Complete and Perfect Information 63 Recall from Chapter 1 that in the Nash equilibrium of the Cournot game each firm produces (a-e)/3. Thus, aggregate quan- tity in the backwards-induction outcome of the Stackelberg game, 3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game. In the Stackelberg gal):1e, how- ever, firm 1 could have chosen its Cournot quantity, (a - e)/3, in which case firm 2 would have responded with its Cournot quan- tity. Thus, in the Stackelberg game, firm 1 could have achieved its Cournot profit level but chose to do otherwise, so firm l's profit in the Stackelberg game must exceed its profit in the Cournot game. But the market-clearing price is lower in the Stackelberg game, so aggregate profits are lower, so the fact tha t firm 1 is better off im- plies that firm 2 is worse off in the Stackelberg than in the Cournot game.

The observation that firm 2 does worse in the Stackelberg than in the Cournot game illustrates an important difference between single- and multi-person decision problems. In single-person deci- sion theory, having more information can never make the decision maker worse off. In game .theory, however, having more informa- tion (or, more precisely, having it known to the other players that one has more information) ean make a player worse off.

In the Stackelberg game, the information in question is firm l 's quantity: firm 2 knows

### q"

and (as importantly) firm 1 knows that firm 2 knows q, . To see the effect this information has, consider the modified sequential-move game in which firm 1 chooses q" after which firm 2 chooses q2 but does so without observing

### q,.

If

firm 2 believes that firm 1 has chosen its Stackelberg quantity

### qi

= (a-e)/2, then firm 2's best response is again R2(qll = (a-e)/4. But if firm 1 anticipates that firm 2 will hold this belief a nd so choose this quaritity, then firm 1 prefers to choose its best response to (a - e)/4-namely, 3(a - e)/B-rather than its Stackelberg quantity (a-e)/2. Thus, firm 2 should not believe that firm 1 h as chosen its Stackelberg quantity. Rather, the unique Nash equilibrium of this

"Stackelberg equilibrium" often mean that the game is sequentia l- rather than simultaneous-move. As noted in the previous section, however, sequential-move games sometimes have multiple Nash equilibria, only one of which is associated with the backwards-induction outcome of the game. Thus, "Stackelberg equilib- rium" can refer both to the sequential-move nature of the game and to the use of a stronger solution concept than simply Nash equilibrium .

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## Stackelberg Model: Remarks

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## Entry Game: Revisit

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## Entry Game and Commitment

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## International Competition 

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### What happens if Airbus has a commitment power?

  Of course, committing to “Produce” is the best.

  In reality, such commitment power is absent / non-credible…

### Don’t Produce 0 100 0 0

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### What if the government can commit to subsidy Airbus?

  Suppose the government pays 10 if Airbus produces.

  Such strategic trade policy might benefit.

  (P, DP) is no longer a Nash equilibrium.

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## Bargaining: Ultimatum Game 

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### subgame, since she can observe the offer of player 1.

  There are 101 subgames. => what is SPNE?

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## Remarks on Ultimatum Game 

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## Extensions of Bargaining Model 

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### Player 1 first names a proposal of shares to player 2.

  The shares realize if Player 2 accepts it; otherwise, player 2 makes a counter proposal to player 1.

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### reaches the final period T. (T could be infinite)

  T = 0: Dictatorship Game (Receiver cannot do anything)

  T = 1: Ultimatum Game

  T = :Alternating Offers Game (Rubinstein Model)

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## Two-Period Model ( T = 2) 

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### There is a unique SPNE (if the offer is continuous):

  Player 1 proposes (1 - δ2

### ,

δ2) in T = 0, and accepts all offers in T

= 2.

  Player 2 accepts any offers weakly greater than

### δ

2 in T =1, and proposes (0, 100) in T = 2.

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## Further Extensions 

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### alternating offer game:

  Increasing the number of periods or players.

  Introducing outside offers, random move, etc.

  Fixed amount of depreciation per period, instead of discounting.

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### shrinks as a common discount factor goes to 1.

  This (almost) equal division is a unique SPNE outcome.

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## Further Exercises 

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### alternating offers game (Rubinstein model), i.e., T = ∞ .

  Show also that the SPNE is unique.

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## 参照

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