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講義 7:動学ゲームと長期的関係

上級ミクロ経済学財務省理論研修

安田 洋祐

政策研究大学院大学(GRIPS)

2012618

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Dynamic Situations | 動学的状況

In a static game (静学ゲーム), players choose strategies simultaneously. Dynamic games (動学ゲーム) take dynamics into account, and consider strategic situations in which players may make choices in sequence.

✂Ex Dynamic Entry Game (✁ 動学的参入ゲーム)

There are two firms in the market game: a potential entrant and a monopoly incumbent. First, the entrant decides whether enter this monopoly market.

1. If the potential entrant (参入企業) stays out, then she gets 0 while the monopolist (独占企業) gets a large profit, say 4.

2. If the entrant enters the market, then the incumbent must choose whether or not to engage in a price war.

3. If price war is triggered, then both firms suffer and receive −1.

4. If he accommodates the entrant, then both firms obtain modest profits, say 1 each.

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Normal-Form Representation | 標準型表現

Each dynamic game can be expressed by a game tree (ゲームの木). (it is formally called extensive-form (展開型) representation)

Dynamic games can also be analyzed in normal-form (標準型): a strategy in dynamic games is a complete action plan which prescribes how the player will act in each possible contingency in the future.

Entrant  Monopolist Price War Accommodation

In −1, −1 1, 1

Out 0, 4 0, 4

There are two Nash equilibria: (In, Acc.) (Out, PW).

✂Rm Dynamic games often have multiple Nash equilibria, and some of them,✁ (Out, PW) in our example, do not seem plausible since they rely on

non-crediblethreats (信憑性の無い脅し).

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Backward Induction | 後方帰納法 (1)

If the entrant chooses “In,” it is optimal for the monopolist to “Accommodate” entry. So, “Price War” is not a credible option. Taking this into account, entrant’s optimal strategy is “In.”

In this way, by solving games from the back to the forward, we can erase those implausible (もっともらしくない) equilibria.

This procedure is called backward induction (後方帰納法).

✂Ex Sequential battle of the sexes, whose simultaneous version is expressed by✁ the following payoff matrix:

Wife  Husband Japanese Italian

Japanese 3, 1 0, 0

Italian 0, 0 1, 3

✂Q What happens if the wife can make decision first (i.e., lady first), and the✁ husband decides after observing wife’s decision?

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Backward Induction | 後方帰納法 (2)

The unique backward induction solution is (Japanese, Japanese).

H will choose the same action as W chose in the 2nd stage.

Taking his optimal reply into account, W will choose the best action, Musical, in the 1st stage.

✂Q Why is extensive-form (solution) better than normal-form?✁

An extensive-form game is an explicit description of the sequential structure of the decision problems encountered by the players in a strategic situation.

It allows us to study solutions in which each player considers her plan of action not only at the beginning of the game but also at any point of time at which she has to make a decision.

By contrast, a normal-form game restricts us to solutions in which each player chooses her plan of action once and for all; the game does not allow a player to reconsider her plan of action after some events in the game have unfolded.

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Translate into Normal-Form | 標準型への変換

Dynamic games can be analyzed in normal-form. Note that a strategy for a player in a dynamic game is a “complete plan” of action: it specifies a feasible action for the player in every contingency in which the player might be called on to act. How can we translate extensive-form representation into normal-form?

Example Sequential battle of the sexes:

W  H J J J I IJ II Japanese 3, 1 3, 1 0, 0 0, 0 Italian 0, 0 1, 3 0, 0 1, 3

In the table, XY means to choose X when W chose J and choose Y when her choise is I, i.e., contingent plan of actions.

✂Rm There are three Nash equilibria, {(J, JJ✁ ), (J, JI), (I, II)} while only (J, JI) is a credible equilibrium.

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Extensive-Form: Definition | 展開型:定義

Def Extensive-form representation of a game specifies

1. the players in the game 2. when each player has the move

3. what each player can do at each of her opportunities to move 4. what each player knows at each of her opportunities to move

5. the payoff received by each player for each combination of moves that could be chosen by the players.

Def An extensive-form game is called a perfect information game (完全情報 ゲーム), if at each move in the game the player with the move knows the full history of the play of the game thus far. If not, the game is called imperfect information game (不完全情報ゲーム).

The next theorem assures that backward induction works for any finite perfect information games.

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Zermelo’s Theorem | ツェルメロの定理

Thm For any finite perfect information games, there exist at least one backward induction solution in pure strategies. Furthermore, if payoffs differ between any two different strategy profiles, there is exactly one backward induction solution.

It establishes the following claim originated by Zermelo (1913).

Thm In any finite two-person perfect information games in which (i) players (1 and 2) have strictly opposing interests (zero-sum game) and (ii) resulting outcomes are either “1 wins,” “2 wins,” or “tie,” then exactly one of the following statements must be true:

1. Player 1 can guarantee that she will win. 2. Player 2 can guarantee that he will win. 3. Each player guarantee at least a tie.

✂Rm The above theorem implies that, from game theoretic point of view, the✁

winner (or result) of chess is determined! 8 / 26

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Game Tree: Definition | ゲームの木:定義

An extensive-form game is defined by a tree that consists of nodes () connected by branches ().

Each branch is an arrow, pointing from one node (predecessor) to another (successor).

A tree starts with the initial node (始点) and ends at terminal nodes (終 点) where payoffs are specified.

Tree Rules

1. Every node is a successor of the initial node.

2. For nodes x, y, and z, if x is a predecessor of y and y is a predecessor of z, then it must be that x is a predecessor of z.

3. The initial node has no predecessor, and every other node has exactly one immediate predecessor.

4. Multiple branches extending from the same node have different action labels.

5. Each information set contains decision nodes for only one of the players.

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Subgame Perfect Equilibrium | 部分ゲーム完全均衡 (1)

Def An information set (情報集合) for a player is a collection of decision nodes satisfying:

the player has the move at every node in the information set

when the play of the game reaches a node in the information set, the player with the move does not know which node has reached.

✂Rm Note that, at every decision node in an information set, each player must✁ choose the same action.

The concept of backward induction can be extended to cover general extensive-form games. We will define a refinement of Nash equilibrium that adequately incorporates sequential rationality (逐次合理性).

Def A subgame (部分ゲーム) in an extensive-form game (1) begins at some decision node n that is a singleton information set, (2) includes all the decision and terminal nodes following n, and (3) does not cut any information sets.

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Subgame Perfect Equilibrium | 部分ゲーム完全均衡 (2)

Def A Nash equilibrium is called a subgame perfect (Nash) equilibrium (部 分ゲーム完全均衡, SPNE) if the players’ strategies constitute a Nash

equilibrium in every subgame.

There are several facts worth noting:

1. Every SPNE is a Nash equilibrium, since such a strategy profile must specify a Nash equilibrium in every subgame by (definition), one of which is the entire game. In other words, SPNE can be seen as a refinement of Nash equilibrium.

2. For games of perfect information, backward induction yields SPNE. 3. To find SPNE, we can work backward by finding NE from the smallest

subgames (toward the end), and move backward.

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Stackelberg Model | シュタッケルベルグ・モデル (1)

The Stackelberg model is a dynamic version of the Cournot model in which one firm (dominant firm) moves first and the other (subordinate firm) moves second.

The game is defined as follows:

Players: Two firms

Strategies: Quantities

Payoffs: Profits

We assume that Firm 1 (a leader) chooses quantity q1 first, and Firm 2 (a follower) observes q1and then chooses its quantity q2.

The profits of the two firms are specified by π1(q1, q2) = q1(1 − q1− q2).

π2(q1, q2) = q2(1 − q1− q2).

Calculation Application 2.1.B (Gibbons, pp.61)

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Stackelberg Model | シュタッケルベルグ・モデル (2)

The point of the Stackelberg model is that commitments matter because of their influence on the rivals’ actions.

The role of the irreversibility of quantity levels (i.e., the fact that they may not be reduced in the future) is important for the investment to have a commitment value.

Firm 1 is not on its best reply curve ex post; its best response to q2= 14 is q1=38 <12.

A leader never becomes worse off since she could have achieved Cournot profit level in the Stackelberg game simply by choosing the Cournot output: a 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 a single-person decision making, having more information can never make the decision maker worse off.

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Finitely Repeated Games | 有限回繰り返しゲーム (1)

A repeated game, a specific class of dynamic game, is a suitable framework for studying the interaction between immediate gains and long-term incentives, and for understanding how a reputation mechanism can support cooperation.

Def Let G = {A1, ..., An; u1, ..., un} denote a static game in which players 1 through n simultaneously choose actions a1 through anfrom the action spaces A1 through An, and payoffs are u1(a1, ..., an) through un(a1, ..., an). The game G will be called the stage game (ステージゲーム) of the repeated game.

Def Given a stage game G, let G(T ) denote the finitely repeated game (有 限回繰り返しゲーム) in which G is played T times, with the outcomes of all preceding plays observed before the next play begins.

Assume that the payoff for G(T ) is simply the sum of the payoffs from the T stage games.

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Finitely Repeated Games | 有限回繰り返しゲーム (2)

Thm If the stage game G has a unique Nash equilibrium, then, for any finite T, the repeated game G(T ) has a unique subgame perfect Nash equilibrium: the Nash equilibrium of G is played in every stage irrespective of the past history of the play.

Proof We can solve the game by backward, that is, starting from the smallest subgame and going backward through the game.

In stage T , players choose a unique Nash equilibrium of G.

Given that, in stage T − 1, players again choose the same Nash

equilibrium outcome, since no matter what they play the last stage game outcome will be unchanged.

This argument carries over backwards through stage 1, which concludes that the unique Nash equilibrium outcome is played in every stage.

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Infinitely Repeated Games | 無限回繰り返しゲーム (1)

Even if the stage game has a unique Nash equilibrium, there may be subgame perfect outcomes of the infinitely repeated game in which no stage game’s outcome is a Nash equilibrium of G.

Def Given a stage game G, let G(∞, δ) denote the infinitely repeated game (無限回繰り返しゲーム) in which G is repeated forever and the players share the discount factor δ.

For each t, the outcomes of the t − 1 preceding plays of the stage game are observed before the t-th stage begins.

Each player’s payoff in G(∞, δ) is the average payoff defined as follows.

Def Given the discount factor δ, the average payoff (平均利得) of the infinite sequence of payoffs π1, π2, ...is

(1 − δ)(π1+ δπ2+ δ2π3+ · · ·) = (1 − δ)

X

t=1

δt−1πt.

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Infinitely Repeated Games | 無限回繰り返しゲーム (2)

There are a few important remarks:

The history (歴史) of play through stage t is the record of the players’ choices in stages 1 through t.

The players might have chosen (as1, ..., asn) in stage s, where for each player ithe action asi belongs to Ai.

In the finitely repeated game G(T ) or the infinitely repeated game G(∞, δ), a player’s strategy specifies the action the player will take in each stage, for every possible history of play.

In the finitely repeated game G(T ), a subgame beginning at stage t + 1 is the repeated game in which G is played T − t times, denoted G(T − t).

In the infinitely repeated game G(∞, δ), each subgame beginning at any stage is identical to the original game.

In a repeated game, a Nash equilibrium is subgame perfect if the players’ strategies constitute a Nash equilibrium in every subgame, i.e., after every possible history of the play.

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Repeated Prisoner’s Dilemma | 繰り返し囚人のジレンマ (1)

✂ ✁

Q The following prisoner’s dilemma will be played infinitely many times. Under what conditions of δ, we can sustain cooperation (C1, C2) as a SPNE?

1  2 D2 C2

D1 1, 1 5, 0

C1 0, 5 4, 4

Suppose that player i plays Ciin the first stage. In the t-th stage, if the outcome of all t − 1 preceding stages has been (C1, C2) then play Ci; otherwise, play Di.

This strategy is called trigger strategy (トリガー戦略), because player i cooperates until someone fails to cooperate, which triggers a switch to noncooperation forever after.

If both players adopt this trigger strategy then the outcome of the infinitely repeated game will be (C1, C2) in every stage.

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Repeated Prisoner’s Dilemma | 繰り返し囚人のジレンマ (2)

To show that the trigger strategy is SPNE, we must verify that the trigger strategies constitute a Nash equilibrium on every subgame of that infinitely repeated game.

✂Rm Since every subgame of an infinitely repeated game is identical to the✁ game as a whole, we have to consider only two types of subgames:

1. subgame in which all the outcomes of earlier stages have been (C1, C2) 2. subgames in which the outcome of at least one earlier stage differs from

(C1, C2).

The next theorem, called Falk theorem, states that large subsets of feasible payoffs are sustained in a Nash equilibrium or in a subgame perfect Nash equilibrium.

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Falk Theorem | フォーク定理

Def We call a combination of payoffs (x1, ..., xn) feasible (達成可能) in the stage game G if it is a convex combination of the pure strategy payoffs of G.

Thm (Falk Theorem) Let G be a finite, static game. Let (e1, ..., en) denote the payoffs from a Nash equilibrium of G, and let (x1, ..., xn) denote any other feasible payoffs from G. If xi> eifor every player i and if δ is sufficiently close to one, then there exists a subgame perfect Nash equilibrium of the infinitely repeated game G(∞, δ) that achieves (x1, ..., xn) as the average payoff.

Proof See Appendix 2.3.B (Gibbons, pp.100)

✂Rm There are a couple of versions of this theorem, and the name comes from✁ the fact that the statement relying on Nash equilibrium as a solution concept was widely known among game theorists in the 1950s, even though no one had published it.

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【補論】 Strategic Investment | 戦略的投資 (1)

Strategic considerations may provide firms an incentive to over-invest in

“capital”to deter the entry or expansion of rivals. The key factors in strategic investment are

whether investment makes the incumbent tough, and

how the entrant reacts to tougher play by the incumbent.

✂Ex A simple two period model of strategic investment✁

In period 1, firm 1 (the incumbent) chooses some variable K1 which we will call an “investment.” Firm 2 observes K1 and decides whether to enter. If it does not enter, it makes zero profit. The incumbent then enjoys a monopoly position in the second period and makes profit

π1m(K1, xm1 (K1)),

where xm1(K1) is the monopoly choice in the second period.

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【補論】 Strategic Investment | 戦略的投資 (2)

If firm 2 enters, the firms make simultaneous second-period choices x1and x2. Them, their profits are written as

π1(K1, x1, x2) and π2(K1, x1, x2).

Suppose that firm 1 chooses K1 and firm 2 enters. Then, the post entry choices x1 and x2 are determined by a Nash equilibrium

{x1(K1), x2(K1)}.

Def We say that entry is deterred if K1 is chosen such that π2(K1, x1(K1), x2(K1)) ≤ 0.

Entry is accommodated if π2(K1, x1(K1), x2(K1)) > 0.

Which of the two cases must be examined depends on whether the incumbent finds it advantageous to deter or to accommodate entry.

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【補論】 Strategic Investment | 戦略的投資 (3)

Let us take the total derivative of π2 with respect to K1, dπ2

dK1

= ∂π2

∂K1

+∂π2

∂x1

dx1

dK1

+∂π2

∂x2

dx2

dK1

= ∂π2

∂K1 direct effect

+ ∂π2

∂x1

dx1

dK1 strategic effect

.

Note that ∂π∂x2

2(K1, x

1(K1), x2(K1)) = 0 by the first order condition (of the firm 2’s optimization problem).

Def We say that investment makes firm 1 tough if dK2

1 <0 and soft if

2 dK1 >0.

To deter entry, firm 1 wants to look tough.

When there is no direct effect (∂K∂π2

1 = 0), “toughness” is fully captured by strategic effect in this entry-deterrence case.

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【補論】 Strategic Investment | 戦略的投資 (4)

When firm 1 decides to accommodate entry, the incentive to invest is given by the total derivative of π1 with respect to K1,

1

dK1

= ∂π1

∂K1 direct effect

+ ∂π1

∂x2

dx2

dK1 strategic effect

.

Under the optimal choice of K1 (given accommodation) dK1

1 = 0 holds, which implies

∂π1

∂K1

<0 ⇔∂π1

∂x2

dx2

dK1

>0 and ∂π1

∂K1

>0 ⇔∂π1

∂x2

dx2

dK1

<0. Now suppose π1 is concave in K1. Then,

Firm 1 over-invests (compared with the case in which there is no strategic effect) if ∂K∂π1

1 <0 and under-invests if ∂K∂π1

1 >0.

In the case of entry accommodation, firm 1 should over-invest (/under-invest) if the strategic effect is positive (/negative).

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【補論】 Strategic Investment | 戦略的投資 (5)

We now further assume that ∂K∂π2

1 = 0, and that the second-period actions of both firms have the same nature, in the sense that∂π∂x1

2 and

∂π2

∂x1 have the same sign. Then,

sign(∂π1

∂x2

dx2

dK1

) = sign(∂π2

∂x1

dx1

dK1

)sign(R2(x1)), since sign(∂π∂x1

2) = sign(

∂π2

∂x1), and

dx2

dK1

=∂x

2

∂x1

dx1

dK1

= R2(x1)dx

1

dK1

.

We are led to distinguish four cases, depending on whether

Investment makes firm 1 tough or soft.

Second-period actions are strategic substitutes (戦略的代替)

(R2(x1) < 0) or strategic complements (戦略的補完) (R2(x1) > 0), i.e., whether best reply curves are downward or upward sloping.

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【補論】 Finitely Repeated Games: Multiple Equilibria

When there are more than one Nash equilibrium in a stage game, multiple subgame perfect Nash equilibria may exist.

Furthermore, an action profile which does not constitute a stage game Nash equilibrium may be sustained (for any period t < T ) in a subgame perfect Nash equilibrium.

✂Q The following stage game will be played twice. Can players sustain✁ non-equilibrium outcome (M1, M2) in the first period?

1  2 L2 M2 R2

L1 1, 1 5, 0 0, 0 M1 0, 5 4, 4 0, 0

R1 0, 0 0, 0 3, 3

✂Rm Note that there are two Nash equilibria in the stage game, i.e., (L✁ 1, L2), (R1, R2): future behavior can influence current behavior in the first period. 26 / 26

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