Incentives and Behavior Prof. Dr. Heiner Schumacher KU Leuven 2. Game Theory II Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 1 / 25
Introduction Until now, we analyzed static games in which players choose their actions simultaneously. We now consider dynamic games in which players decide sequentially. The only new element that we add is a time structure. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 2 / 25
Introduction As an example, we consider a game with two players. The second party observes the …rst party’s choice and then chooses its strategy. We can display such games graphically in a game tree: Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 3 / 25
Introduction The players have the following strategies: Player A has the strategies L and R . Player B has the strategies ll , lr , rl und rr . Interpretation of lr : “Play l , if A chose strategy L , and r , if A chose strategy R ”. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 4 / 25
Introduction In a sequential game, a strategy speci…es for each decision node the action that will be chosen if the decision node is reached. How can we solve a sequential game? One option is to translate it back into its “normal form” (the payo¤ matrix that we used for static games): A / B ll lr rl rr L 1,9 1,9 3,8 3,8 R 0,0 2,1 0,0 2,1 Show that ( R , lr ) and ( L , ll ) are Nash equilibria. Are these convincing outcomes? Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 5 / 25
Introduction The equilibrium ( L , ll ) is not very plausible: In this equilibrium, player B threatens to choose l if A chooses R . If A believes that B realizes her threat, it is optimal for A to choose L . However, the threat is not credible. If A chooses R , it would be better for B to choose r ! If A anticipates that B will choose r , it is better for her to choose R instead of L . Show that this problem does not occur at the equilibrium ( R , lr ) ! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 6 / 25
Introduction Overview De…nitions Subgame-Perfect Equilibrium Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 7 / 25
De…nitions The extensive form representation of games contains the following information: The set of players f 1 , ..., n g . 1 The order of moves, i.e., which player moves when. 2 What the players possible actions are when they move. 3 What each player knows when he makes his decision. 4 The players’ payo¤s as a function of moves that were made. 5 The probability distribution over any exogenous events. 6 Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 8 / 25
De…nitions Point 2 and 3: How can we describe such games? If the game is su¢ciently simple, we use a game tree (as in the example above). A game tree is a set of ordered and connected nodes. The game starts with exactly one initial node . Decision nodes : Exactly one player can choose an action. Each action leads to a new decision or terminal node. Terminal nodes : Here the game ends and payo¤s are realized. This structure excludes cycles! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 9 / 25
De…nitions Point 4: How can we describe what a player knows when he moves? We use information sets in order to describe the informational structure of the game. An information set for a player is a set of decision nodes satisfying the following properties: (1) At all nodes of the information set, the same player moves; (2) The player cannot distinguish between di¤erent nodes of the information set; this implies that at every node of the information set, she faces the same set of actions. Every node is an element of exactly one information set. Players always recall their own actions (unless speci…ed otherwise)! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 10 / 25
De…nitions If all information sets of a game are singletons, we have a game with perfect information (e.g., the game from the introduction). If at least one information set contains more than one node, we have a game with imperfect information . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 11 / 25
De…nitions CAUTION . We must not confuse perfect/imperfect information with complete/incomplete information. In games of incomplete information, players may not know the opponents’ preferences. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 12 / 25
De…nitions Point 6: Exogenous events may be part of the game. We model exogenous events as moves by the player “nature”. Nature chooses events from a given set of events according to some probability distribution (of course, nature is not a real player and earns no payo¤s). In the game on the next slide, there are two real players, Z and M . After Z has chosen her action, nature (denoted as “Natur”) chooses between “left” and “right” with equal probability. Then player M chooses between k and n . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 13 / 25
De…nitions Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 14 / 25
De…nitions The strategy space in dynamic games is more complex than in static games. A strategy for a player is a complete plan of action: it speci…es a feasible action for the player in every information set in which the player might be called to act. Note the di¤erence between nodes and information sets! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 15 / 25
De…nitions What are the strategies of player 1? What are the strategies of player 2? Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 16 / 25
De…nitions Consider the game from the last slide. Suppose now the second player cannot observe the action of the …rst player. How do you have to adjust the game tree? What are the strategies of player 1? What are the strategies of player 2? Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 17 / 25
De…nitions What are the strategies of player 1? What are the strategies of player 2? Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 18 / 25
Subgame-Perfect Equilibrium As we saw in the example of the introduction, the Nash equilibrium ignores the dynamic structure of an extensive game: Nash equilibria may rely on non-credible threats. If we want to exclude implausible threats, we have to re…ne our equilibrium concept. Each player’s strategy must be optimal, given the other players’ strategies not only at the start of the game, but in each “subgame” – even if it is not reached in equilibrium. This is the idea behind Reinhart Selten’s (1965) “subgame-perfect equilibrium”. 1 1 Selten, Reinhard (1965): “Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit – Teil I: Bestimmung des dynamischen Preisgleichgewichts,” Zeitschrift für die gesamte Staatswissenschaft 121, 301-324. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 19 / 25
Subgame-Perfect Equilibrium A subgame in an extensive form game has the following properties: It begins at a decision node α that is a singleton information set. 1 It includes all the decision and terminal notes following α in the game 2 (but no nodes that do not follow α ). It does not cut any information sets. 3 Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 20 / 25
Subgame-Perfect Equilibrium Less formal: A subgame is a part of the game that remains to be played starting at a point at which the complete history of the game thus far is common knowledge among all players. Each extensive form game has at least one subgame (i.e., the game itself). What are the subgames in the extensive from games displayed on slide 16 and 18? Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 21 / 25
Subgame-Perfect Equilibrium De…nition of a Subgame-Perfect Nash Equilibrium. A Nash equilibrium is subgame-perfect if the players’ strategies constitute a Nash equilibrium in every subgame. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 22 / 25
Subgame-Perfect Equilibrium We …nd a subgame-perfect equilibrium through backward-induction . Find the optimal actions of the player who chooses last. Substitute the last decision nodes by the payo¤s that would occur if the last player chooses an optimal action in each subgame. Consider the penultimate decision nodes and proceed in the same manner. Continue until you reach the …rst decision node. Apply this procedure to the game from the introduction! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 2. Game Theory II 23 / 25
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