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 .
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. R2(q,) solves
max 7r2(q" q2) = max q2[a - q, - q2 - eJ,
Q2?:O
which yields
a -
q, -
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 towhich yields
qi
= -2-a-e and R2(qi) = -4-a-eas 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 knowsq, .
To see the effect this information has, consider the modified sequential-move game in which firm 1 choosesq"
after which firm 2 chooses q2 but does so without observing
q,.
Iffirm 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. R2(q,) solves
max 7r2(q" q2) = max q2[a - q, - q2 - eJ,
Q2?:O
which yields
a -
q, -
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 towhich 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 observingq,.
Iffirm 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)
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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 - 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.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.
