The Extensive Form Backward Induction Subgame Perfect Equilibrium

Lecture 24 Extensive Form Games

Jitesh H. Panchal

ME 597: Decision Making for Engineering Systems Design Design Engineering Lab @ Purdue (DELP) School of Mechanical Engineering Purdue University, West Lafayette, IN http://engineering.purdue.edu/delp

November 14, 2019

ME 597: Fall 2019 Lecture 24 1 / 31 The Extensive Form Backward Induction Subgame Perfect Equilibrium Lecture Outline

1 The Extensive Form

2 Backward Induction Examples

3 Subgame Perfect Equilibrium Definition Examples

Dutta, P.K. (1999). Strategies and Games: Theory and Practice. Cambridge, MA, The MIT Press. Chapters 11-13. ME 597: Fall 2019 Lecture 24 2 / 31 The Extensive Form Backward Induction Subgame Perfect Equilibrium Game Tree

A Game Tree Decision Node: Point in the game where one player–and only one player–has to make a decision. Branch: Each branch corresponds to one of the choices. Root: Starting point of the tree. Terminal node: No branches emanating from it.

Figure: 11.1 on Page 158 (Dutta) ME 597: Fall 2019 Lecture 24 3 / 31 The Extensive Form Backward Induction Subgame Perfect Equilibrium Game tree

Information set (oval) represents simultaneous moves. It represents nodes that are indistinguishable from the decision maker’s standpoint.

Figure: 11.2 on Page 159 (Dutta)

ME 597: Fall 2019 Lecture 24 4 / 31 The Extensive Form Backward Induction Subgame Perfect Equilibrium Consistency Requirements

1 Single starting point 2 No cycles 3 One way to proceed: There must not be two or more branches leading to a node.

ME 597: Fall 2019 Lecture 24 5 / 31 The Extensive Form Backward Induction Subgame Perfect Equilibrium Strategies

A player’s strategy is a complete, conditional plan of action.

Complete: what to choose at every relevant decision node. Conditional: which branch to follow out of a decision node if the game arrives at that node.

ME 597: Fall 2019 Lecture 24 6 / 31 The Extensive Form Backward Induction Subgame Perfect Equilibrium Mixed Strategies

Same as in strategic form: a probability distribution over the pure strategies.

ME 597: Fall 2019 Lecture 24 7 / 31 The Extensive Form Backward Induction Subgame Perfect Equilibrium Chance Nodes

Chance node: Nodes whose branches represent several random possibilities. Used to represent uncertainty inherent in the game (as opposed to uncertainty introduced by players through mixed strategies.)

Figure: 11.5 on Page 162 (Dutta)

ME 597: Fall 2019 Lecture 24 8 / 31 The Extensive Form Backward Induction Subgame Perfect Equilibrium Perfect Information Games

Definition (Perfect Information Game) A game of perfect information is one in which there is no information set (with multiple nodes).

Any time a player has to move, he/she knows exactly the entire history of choices made by all previous players. Game of perfect information cannot have any simultaneous moves.

ME 597: Fall 2019 Lecture 24 9 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Basic Idea: Sequential Rationality

Rationality: Each player picks the best action available to him at a decision node, given what he thinks is going to be the future play of the game.

Sequential: Players will infer what the future is going to be knowing that, in the future, players will reason in the same way.

ME 597: Fall 2019 Lecture 24 10 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Example - Entry Game

Every strategy has three components (e.g., EAT).

Figure: 11.7 on Page 164 (Dutta)

ME 597: Fall 2019 Lecture 24 11 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Example – Entry Game (contd.)

Strategic form: Coke / Pepsi TA ETT −2, −1 0, −3 ETA −2, −1 1, 2 EAT −3, 1 0, −3 EAA −3, 1 1, 2 OTT 0, 5 0, 5 OTA 0, 5 0, 5 OAT 0, 5 0, 5 OAA 0, 5 0, 5

How many pure strategy Nash equilibria does this game have?

ME 597: Fall 2019 Lecture 24 12 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Pure Strategy Nash Equilibria

1 Pepsi: T ; Coke: OTT , OTA, OAT , or OAA 2 Pepsi: A; Coke: ETA 3 Pepsi: A; Coke: EAA

The only sequentially rational strategy is: Pepsi: A; Coke: ETA

ME 597: Fall 2019 Lecture 24 13 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Backward Induction: General Result

Kuhn’s Theorem Every game of perfect information with a finite number of nodes has a solution to backward induction. Indeed, if for every player it is the case that no two payoffs are the same, then there is a unique solution to backward induction.

Fold the decision tree back one step at a time till we reach the beginning.

Proof by Induction...

ME 597: Fall 2019 Lecture 24 14 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Backward Induction: Equivalent to IEDS in the strategic form

Example: Coke / Pepsi TA ETT −2, −1 0, −3 ETA −2, −1 1, 2 EAT −3, 1 0, −3 EAA −3, 1 1, 2 OTT 0, 5 0, 5 OTA 0, 5 0, 5 OAT 0, 5 0, 5 OAA 0, 5 0, 5

ME 597: Fall 2019 Lecture 24 15 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Another Example

Figure: 11.13 on Page 173 (Dutta)

ME 597: Fall 2019 Lecture 24 16 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Example: Research and Development

Questions of interest: How much should each firm spend on R&D, and how often? When should it get into the race, and at what point should it opt out of such a race? What factors determine the likely winner: is it an advantage to be in a related manufacturing area, is it more important to have a superior R&D department, and so on?

ME 597: Fall 2019 Lecture 24 17 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Example: A Model of Research and Development

Suppose there are two firms: RCA (R) and Sony (S)

Simplifying assumptions: 1 The distance from the eventual goal can be measured; we can say, for example, that firm S is n steps from completing the project. 2 Either firm can move 1, 2, or 3 steps closer to the end in any one period. 3 It costs $2 to move one step forward, $7 to move two steps forward, and $15 to move three steps forward. 4 Whichever firm completes all the steps first gets the patent; the patent is worth $20. 5 The firms take turns deciding how much to spend on R&D; if RCA makes an R&D decision this period, it waits to make any further decisions till it learns of Sony’s next R&D commitment. Furthermore, Sony makes its announcement in the period following RCA’s announcement.

ME 597: Fall 2019 Lecture 24 18 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Example: A Model of Research and Development

Figure: 12.1 on Page 183 (Dutta)

ME 597: Fall 2019 Lecture 24 19 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Example: A Model of Research and Development

Figure: 12.2 on Page 184 (Dutta)

ME 597: Fall 2019 Lecture 24 20 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Example: A Model of Research and Development

Figure: 12.3 on Page 185 (Dutta)

ME 597: Fall 2019 Lecture 24 21 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Example: A Model of Research and Development

Figure: 12.4 on Page 186 (Dutta)

ME 597: Fall 2019 Lecture 24 22 / 31 The Extensive Form Backward Induction Examples Subgame Perfect Equilibrium Example: A Model of Research and Development

Figure: 12.5 on Page 187 (Dutta)

ME 597: Fall 2019 Lecture 24 23 / 31 The Extensive Form Definition Backward Induction Examples Subgame Perfect Equilibrium Subgames

Game of Imperfect Information: Game in extensive form that is not a game of perfect information.

Definition (Subgame) A subgame is a part of the extensive form: it is a collection of nodes and branches that satisfies three properties: 1 It starts at a single decision node. 2 It contains every successor to this node. 3 If it contains any part of an information set, then it contains all the nodes in that information set.

Once you are in a subgame, you cannot stray out of it. You will have all the information within the subgame to make decisions.

ME 597: Fall 2019 Lecture 24 24 / 31 The Extensive Form Definition Backward Induction Examples Subgame Perfect Equilibrium Subgames – Examples

Not sub-games:

Figure: 13.2-13.4 on Page 197 (Dutta)

ME 597: Fall 2019 Lecture 24 25 / 31 The Extensive Form Definition Backward Induction Examples Subgame Perfect Equilibrium Subgame Perfect Equilibrium

s1 and s2 constitute a subgame perfect (Nash) equilibrium if for every subgame g, s1(g) and s2(g) constitute a Nash equilibrium within the subgame.

ME 597: Fall 2019 Lecture 24 26 / 31 The Extensive Form Definition Backward Induction Examples Subgame Perfect Equilibrium Subgame Perfect Equilibrium

In a game of perfect information, the subgame perfect equilibria are precisely the backward induction solutions. Hence, whenever the backward induction solution is unique, there is a single subgame perfect equilibrium.

ME 597: Fall 2019 Lecture 24 27 / 31 The Extensive Form Definition Backward Induction Examples Subgame Perfect Equilibrium Subgame Perfect Equilibrium – Example

Once repeated Prisoner’s Dilemma

ME 597: Fall 2019 Figure: 13.5 on PageLecture 199 24 (Dutta) 28 / 31 The Extensive Form Definition Backward Induction Examples Subgame Perfect Equilibrium Subgame Perfect Equilibrium – Example

Voting

Figure: 13.6 on Page 201 (Dutta)

ME 597: Fall 2019 Lecture 24 29 / 31 The Extensive Form Definition Backward Induction Examples Subgame Perfect Equilibrium Summary

1 The Extensive Form

2 Backward Induction Examples

3 Subgame Perfect Equilibrium Definition Examples

ME 597: Fall 2019 Lecture 24 30 / 31 The Extensive Form Definition Backward Induction Examples Subgame Perfect Equilibrium References

1 Dutta, P.K. (1999). Strategies and Games: Theory and Practice. Cambridge, MA, The MIT Press. Chapters 11-13.

ME 597: Fall 2019 Lecture 24 31 / 31