Dynamic Games & The Extensive Form

1. Dynamic Games & The Extensive Form In simultaneous move games, all players move at the same time once - no player observes the others’ moves before determining his own strategy In dynamic games - players may move sequentially or move many times - there may be multiple stages, multiple periods, and repetition - history can matter - the info available to each player at each point in the game must be described Game trees are often useful for represented an extensive form game Example: A potential entrant decides whether to enter a market currently occupied by a monopolist The monopolist observes the potential entrant’s choice and decides whether to start a price war to drive the entrant out of business

2. Potential entrant The game tree summarizes the order of moves, strategies and payoffs. We need to describe a subgame A subgame starts at a node, and all the players recognize that they are at a particular node. Also, once players begin playing a subgame they continue to play it for the rest of the game Essentially the subgame is like a game itself but it is within the larger game Different information conditions: You could have a situation where the 2nd player doesn’t observe the first player’s action but needs to react anyway

3. Military situation What happens when communications are down? U.S. does not observe whether them has launched or not and must decide whether to retaliate or not. In other words, U.S. does not know which node it is at So the only subgame is the entire game An information set is a collection of nodes -a player has the same action choices at each node in a given information set -a player doesn’t know which node he is at unless the info set only has one node

4. Information Set A subgame is a subpart of the overall game - it begins at a node, all players know that they are at the node and once the subgame begins, players remain in it for the rest of the game Information Set - elements are a subset of a particular player’s decision nodes - the reason the player cannot distinguish the nodes is that the player does not observe something about what previously transpired in the game - at every node in the info set, a player must face the same possible actions A listing of all a player’s information sets - “from the player’s perspective” all of the possible distinguishable events or circumstances in which the player might be called upon to move

5. Credibility The credibility of a player’s strategy becomes an issue in dynamic games. We could always just analyze extensive form games using the normal form representation, and continue to rely on Nash equilibrium to predict outcomes and analyze strategies, but we will see that NE does not rule out some non-credible threats Problems with NE: A) allows M’s empty threats to influence E’s behavior- if E is not influenced, M will not act on the threat B) actions at decision nodes that are unreached by play of the equilibrium strategies do not affect M’s payoff - so M can plan to do anything Defn: A strategy profile is a subgame perfect Nash equilibrium of a game if the strategy profile is a Nash equilibrium for every subgame. - this is the first equilibrium refinement; the SPNE is more restrictive than the NE - players must best respond to each other in each subgame

6. Subgame Perfect Nash Equilibrium To see the distinction, consider the game with the monopolist and the entrant Claim: One Nash Equilibrium is as follows: E: stay out M: cut price if E enters It is straightforward to check that each player is best responding to the other - E knows that if M cuts price if E enters then E should stay out so given M’s strategy E is best responding. - M knows that if E stays out, M does not have to implement the price cutting M is best responding The Key: M can cut price if E enters because E never enters in equilibrium so M does not need to implement that part of the strategy The SPNE restriction requires - M plays optimally, even in subgames that are never reached - In the unique SPNE of this game, M accommodates E

7. The Path of Play and NE vs. SPNE The equilibrium path of play is the path players follow on the game tree when they play their equilibrium strategies This is distinct from a strategy profile Strategies are compete contingent plans; the path keeps track of only the part of the strategy that is implemented in equilibrium The Path: The sequence of actions chosen in equilibrium Off the Path: Decision nodes that are unreached by play of the equilibrium strategies In a subgame perfect Nash equilibrium, players must best respond on and off the path In a Nash equilibrium, players are not required to best respond off-the-path

8. Perfect Information Defn: A game is one of perfect information, if each information set contains a single decision node. Otherwise, it is a game of imperfect information. Backward induction can be used to solve for the SPNE of any finite game of perfect info -start at the end of the game with all final subgames -compute best responses and payoffs - work you way back up the tree - it’s critical to have a place to start; the game must be finite The perfect information requirement requires: - at every point in the game only one player moves - player knows the entire history of actions up to that point Prop: Every finite game of perfect information has a pure strategy SPNE. Moreover, if no player has the same payoffs at any two terminal nodes, then there is a unique subgame perfect Nash equilibrium.

9. The Centipede Game 2 players take turns Si= {stop, continue} As soon as one player’s play stops, the game ends and payoffs are realized. The extensive form: The pot keeps growing but the distribution of gains changes from turn to turn -if players let the pot grow until the end then each gets 100 Unique SPNE: Each player stops at every node. The SPNE payoffs are {1,1} This calls the SPNE concept into question - clearly player (1) should only stop when payoffs are {1, 1} if (2) stops at {0,3} - the assumption that both players play SPNE strategies is important - if player (2) isn’t rational (1) could do better

10. Unusual analysis The analysis of nodes after the first is unusual in the Centipede game The unique SPNE is for each player to stop at every node so if a player ever is called upon to play after the first node then that player could conclude that a SPNE is not being played Therefore, the player’s BR might be something other than stop Why would the player assume that the remaining subgame would be played according to the SPNE strategies if the game up to that point has not been played with those strategies? SPNE – most useful for models with - a small number of stages - a simple information structure - incentives that dominate norms or customs - optimizing players: (profit seeking) firms vs. (norm-influenced) individuals

11. Generalized backward induction • We can go further and consider games of imperfect information to identify the set of subgame perfect Nash equilibrium in any finite dynamic game • Start at the end of the game tree and identify the Nash equilibria for each of the final subgames (final subgames have no other subgames nested within them) • Select one Nash equilibrium in each of these final subgames and derive the reduced extensive form game in which final subgames are replaced by the payoffs that result when players use these equilibrium strategies • Repeat steps (1) and (2) for the reduced game and continue until every move is determined. The resulting strategy profile is a SPNE • If multiple equilibrium are never encountered at any step, then this strategy profile is the unique SPNE. Otherwise, repeating the procedure for each possible equilibrium that could occur in every subgame identifies the full set of SPNE • In the case of perfect information, this reduces to simple backward induction

12. The Market Niche Game Entrant decides whether to enter a market occupied by a Monopolist. If E enters, E & M, simultaneously choose which niche to occupy: There is a small and a large niche Following generalized backward induction, suppose E enters, then E and M play the simultaneous move game We’ll just consider pure strategy equilibria There are two: (1) E small M large (2) E large M small

13. Reduced extensive form game Equilibrium strategies (A) E small; M large (B) E large; M small With equilibrium strategy (A) the reduced extensive form game becomes: and E optimally stays out So one SPNE is E out, small niche if in M large niche if E plays in With equilibrium strategy (B) the reduced extensive form game becomes: and E optimally enters So another SPNE is E in, large niche if in M small niche if E plays in