Lecture 8: Dynamic Games
Advanced Microeconomics II
Yosuke YASUDA
National Graduate Institute for Policy Studies
January 14, 2014
Dynamic Situations
In a static game, players choose strategies simultaneously. Dynamic games take dynamics into account, and consider
strategic situations in which players may make choices in sequence.
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Ex Dynamic Entry Game✆
There are two firms in the market game: a potential entrant and a monopoly incumbent. First, the entrant decides whether or not to enter this monopoly market.
1. If the potential entrant stays out, then she gets 0 while the monopolist 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
Each dynamic game can be expressed by a game tree. (it is formally called extensive-formrepresentation)
Dynamic games can also be analyzed in normal-form: 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).
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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-credible threats.
Backward Induction (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 equilibria.
◮ This procedure is called backward induction.
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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
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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 (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.
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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.
Translate into Normal-Form
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 Payoff matrix becomes as follows:
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
where XY′ means to choose X when W chose J and choose Y when her choise is I.
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
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, 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.
The next theorem assures that backward induction works for any finite perfect information games.
Zermelo’s Theorem
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.
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Rm The above theorem implies that, from game theoretic point✆ of view, the winner (or result) of chess is determined!
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Game Tree: Definition
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 and ends at terminal nodes 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.
Subgame Perfect Equilibrium (1)
Def An information set for a player is a collection of decision nodes satisfying: (1) the player has the move at every node in the information set, and (2) when the play of the game reaches a node in the information set, the player with the move does not know which node in the information set has reached.
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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.
Def A subgame 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 other information sets.
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Subgame Perfect Equilibrium (2)
Def A Nash equilibrium is called a subgame perfect (Nash) equilibrium (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.
Stackelberg Model (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 q1 and 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 (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 <
1 2.
◮ 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.
