## ECO290E: Game Theory

Lecture 9: Applications of Dynamic Games

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## Review of Lecture 9

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### An extensive-form game is expressed by a tree,

### which consists of nodes connected by branches.

### A strategy in a dynamic game is a complete plan

### of actions, i.e., specifying what she will do in every

### her information set.

### The subgame perfect Nash equilibrium is a

### combination of dynamic strategies that constitute

### a Nash equilibrium in every subgame.

⇒_{ }

### SPNE is stronger solution concept than NE.

## Cournot Model: Revisit

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### Players: Two firms

### Strategies: Quantities they will charge

### Payoffs: Profits

### Assumptions:

### A linear demand function: *P = a - bQ *

### Common marginal cost, ^{c} .

^{c}

### Firms cannot decide their prices to charge, but the

### market price is determined so as to clear the market.

⇒_{ }

### See Handout (“Oligopoly Competition”).

## Stackelberg Model

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### The Stackelberg model is a dynamic (sequential-

### move) version of the Cournot model in which a

### dominant firm (leader) moves first and a subordinate

### firm (follower) moves next.

### Firm 1 (a leader) chooses a quantity first.

### Firm 2 (a follower) observes firm 1’s quantity and then

### chooses its own quantity.

⇒_{ }

* Let b = 1, and solve the game backwards! *

## 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.
*R*_{2}*(q,) * solves

*max 7r2(q" q2) * ^{= } *max q2[a - q, - q2 - eJ, *

**Q2?:O **

which yields

*a *-

*q, * -

e
*R*

^{2}

^{(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 R*

_{2}*(q,}*is truly firm 2's reac- tion to firm l's observed quantity, whereas in the Cournot analYSis

*R*

_{2}*(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, . }

*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 . **

## 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.
*R*_{2}*(q,) * solves

*max 7r2(q" q2) * ^{= } *max q2[a - q, - q2 - eJ, *

**Q2?:O **

which yields

*a *-

*q, * -

e
*R*

^{2}*(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 R*

_{2}*(q,}*is truly firm 2's reac- tion to firm l's observed quantity, whereas in the Cournot analYSis

*R*

_{2}*(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 }

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

_{q, . }*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 . **

## Stackelberg Model: Remarks

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### Compared to the Cournot outcome, a leader never

### becomes worse off!

### since she could have achieved Cournot profit level in the

### Stackelberg game simply by choosing the Cournot output.

⇒_{ }

### Gain from commitment.

### A follower does become worse off although he has

### more information in the Stackelberg game than in the

### Cournot game, i.e., the rivals output.

### Note that, in single-person decision making, having more

### information can never make the decision maker worse off.

## Entry Game: Revisit

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### What if the monopolist can commit to price war?

**(0,4)** ^{
}

**(-1,-1)** ^{
}

**(1,1)** ^{
}

**Entrant** ^{
}

**Monopolist** ^{
}

**OUT** ^{
}

**IN** ^{
}

**WAR** ^{
}

**ACC.** ^{
}

## Entry Game and Commitment

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### Committing NOT to take ex-post optimal strategy

### (Accommodation) increases monopolist’s payoff.

**(0,4)** ^{
}

**(-1,-1)** ^{
}

**(1,1)** ^{
}

**Entrant** ^{
}

**Monopolist** ^{
}

**OUT** ^{
}

**IN** ^{
}

**WAR** ^{
}

**ACC.** ^{
}

## International Competition ^{}

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### There are two asymmetric Nash equilibria.

### 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…

**Boeing ** ^{╲} ** Airbus** ^{
} **Produce** ^{
} **Don’t Produce** ^{
}

**Produce** ^{
} ** -5 -5** ^{
} ** 100 0** ^{
}

**Don’t Produce** ^{
} ** 0 100** ^{
} ** 0 0** ^{
}

## Strategic Trade Policy ^{}

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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.^{
}

**Boeing ** ^{╲} ** Airbus** ^{
} **Produce** ^{
} **Don’t Produce** ^{
}

**Produce** ^{
} ** -5 5** ^{
} ** 100 0** ^{
}

**Don’t Produce** ^{
} ** 0 110** ^{
} ** 0 0** ^{
}

## Bargaining: Ultimatum Game ^{}

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### Splitting $100 between two players.

### Player 1 makes a take-it-or-leave-it offer to Player 2.

### If player 2 rejects the offer, both players receive 0.

### The ultimatum bargaining game is perhaps the simplest of

### dynamic bargaining models.

### Note that, every decision node for player 2 initiates a

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

There are 101 subgames. => what is SPNE?^{
}

## Remarks on Ultimatum Game ^{}

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### In the ultimatum game, there are two SPNE:

### Player 1 offers (100, 0) and player 2 accepts all offers.

### Player 1 offers (99, 1) and 2 accepts all offers but (100, 0).

### The proposer (player 1) can take almost all the surplus.

### Huge bargaining power in position of making a take-it-or-

### leave-it offer .

### However, the result is extremely unequal.

### Experimental results typically contradict to this outcome.

### The actual bargaining can be much more complicated.

## 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.

### The game continues until some proposal is accepted, or it

*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)

### Both player dislikes delay: future payoffs are discounted by

### some common discount factor, δ

^{1}

### , δ

^{2}

### . ^{
}

## Two-Period Model ( _{T} = 2) ^{}

_{T}

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### Two-period alternating offer game consists of repetition

### of the ultimatum game structure.

### In the second period, player 2 can get entire surplus, so

### player 1 should take this into account when he initially

### makes an offer.

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

Player 1 proposes (1 - ^{δ}^{2}

_{, }

^{δ}

^{2}) 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.

## Further Extensions ^{}

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### There are many ways to extend our basic two-period

### 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.

### All of these new elements affect bargaining outcomes,

### providing better understanding of bargaining process.

### In the infinite period model (Rubinstein model), the first

### proposer (player 1) can still get some advantage, but her gain

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

This (almost) equal division is a unique SPNE outcome.^{
}

## Further Exercises ^{}

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### Find a real life example of commitment.

### Derive a subgame perfect Nash equilibrium in our

### dynamic game with T = n.

### Derive a subgame perfect Nash equilibrium in the

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

Show also that the SPNE is unique.^{
}