Multistage Games

Introduction

Normal-form games models a game where players choose their moves simultaneously without observing other players moves. Extensive-form games adds a possibility playing sequentially - allowing players to learn about the choices of previous players, so the player can condition his moves based on previous players decisions. Extensive-form games have their payoff expressed after the end of the game (one "grand game"). In reality, dynamic play over time may be more complex than one game that unfolds over time. Instead, players can play one game that is followed by another, or maybe even several other games.

• Should we consider each game independently, or should we expect players to consider the sequence of different games as one "grand game"?
• How to evaluate total payoffs from a sequence of payoffs in each of the sequentially played stage-games? What is the value of payoff in period N and in period N+20? If it was just a one-stage-game we could "solve" the game using Weak Perfect Bayesian Equilibrium...
• Will the players moves vary if the stage-game were not played sequentially but as independent games?
• Will a player vary from the stage-game Nash equilibrium to a different action, which will result him a higher payoff in the following stage-games?

Definitions

Multi-stage game

A definition of multi-stage game by S. Tadelis is following: A multi-stage game is a finite sequence of stage-games, each one being a game of complete but imperfect information (a simultaneous move game). These games are played sequentially by the same players, and the total payoffs from the sequence of games will be evaluated using the sequence of outcomes in the games that were played. We adopt the convention that each game is played in a distinct period, so that game 1 is played in period 1, game 2 in period 2, and so on. We will also assume that after each stage is completed, all the players observe the outcome of that stage, and that this information structure is common knowledge.

Multi-stage game consists of multiple games. In each game, players have a set of action they choose from, and the profiles of actions lead to payoffs for that specific game, which is then followed by another game and another, until sequence of games is over. Once we realize that one game follows another, this implies that players can observe the outcomes of each game before another game is played. This observation is important because it allows players to condition their future actions on past outcomes. This is the idea at the center of multi-stage games: the ability to condition behavior may lead to a rich set of outcomes. In what follows, we will analyze the idea of conditional strategies, and the equilibria that can be supported using such strategies.

Information Set

Information set (dashed blue line)

Information Set is collection of decision nodes such that:

• When the play reaches a node in the information set, the player with the move does not know which node in the information set has been reached.
• The player has the same set of choices at each node in the information set.

Information Set can be used in extensive-form games to view a subgame with simultaneous moves of the players. If the player doesn't have the same set of choices at each node it is not an Information set. The point is that for the player its unknown and undecidable know at what node we are. Why is Information Set used? Because it allows us to view a complicated multi-stage game in a familiar tree view, which is also used in Dynamic Games.

Solved Example

(more examples are better; one simple, others more advanced)

Relevant facts

(how the phenomenon is used, etc.)

Exercises

that students could use to practice the problem and that could be an inspiration for test questions