Backward Induction Economics 302 - Microeconomic Theory II: Strategic Behavior

Shih En Lu

Simon Fraser University (with thanks to Anke Kessler)

ECON 302 (SFU) Backward Induction 1 / 15 Topics

1 Introduction to Sequential (also known as Dynamic) Games 2 Backward Induction and -Perfect Equilibrium

ECON 302 (SFU) Backward Induction 2 / 15 Most Important Things to Learn

1 Basic concepts: extensive form (), strategies in sequential games, 2 Know why NE may not work well in sequential games 3 Definition and motivation for subgame-perfect equilibrium (SPE) 4 Using backward induction, and knowing when it yields a unique solution 5 Know the difference between SPE and NE

ECON 302 (SFU) Backward Induction 3 / 15 Introduction to Sequential Games

Up to now: studied games where players move simultaneously. But often, people/firms observe what others do before acting. Does it make a difference? Example: Battle of the Sexes Brett Choir Basketball If simultaneous move: Amy Choir 6,4 1,1 Basketball 0,0 3,7 What happens if Amy texts Brett: "I’mgoing to the choir, and my phone is dying. See you there!" So if Amy has the opportunity to send that text (and, for whatever reason, Brett doesn’t),will she do it?

ECON 302 (SFU) Backward Induction 4 / 15 The Extensive Form

A good way to represent a is the extensive form, often called a "game tree." Consider the Battle of the Sexes, with Amy (player 1) first texting Brett "Choir" (C) or "Basketball" (B), and then becoming out of reach. Each branch is an action. Let’scall Brett’s(player 2’s)actions C’and B’to distinguish them from Amy’s. Each non-terminal node is a place where the specified player has to make a decision. Each terminal node is an : a combination of actions, just like before. The numbers below each terminal node are the payoffs from the outcome corresponding to the node. As usual, the first number is player 1’spayoff, the second is player 2’s,etc.

ECON 302 (SFU) Backward Induction 5 / 15 Perfect Information

We will first study games of perfect information: one player acts at a time, and each player sees all previous actions. Simultaneous-move games are NOT games of perfect information (when at least two players have at least two actions each). Then, we will look at games that do not have perfect information. Example of the latter: playing a prisoner’sdilemma more than once. Note: don’tconfuse perfect information with !

ECON 302 (SFU) Backward Induction 6 / 15 Strategies in Sequential Games

A player’s specifies a probability distribution over her actions at each node where she plays, regardless of whether that node is reached. In other words, a strategy is a player’s full contingency plan. In our example, Brett’sstrategy must include what he would do if Amy chooses B, even if we don’texpect Amy to choose B. Example 1: Brett chooses B’regardless of what Amy does. (C B’,B B’) → → Example 2: Brett chooses the same thing as the Amy. (C C’,B B’) → → Just like before, a strategy profile is a collection of each player’s strategy.

ECON 302 (SFU) Backward Induction 7 / 15 in Sequential Games

Let’sfind the pure-strategy NE in this game. As before, we use the normal form: (C C’,B C’) (C C’,B B’) (C B’,B C’) (C B’,B B’) C→ 6,4→ → 6,4→ → 1,1→ → 1,1 → B 0,0 3,7 0,0 3,7 Are these pure-strategy NE all realistic?

ECON 302 (SFU) Backward Induction 8 / 15 Subgame Perfection

Idea: should require that players play a best-response (given what they know) at all nodes, even those that are not reached. Strategy profiles satisfying the above are called subgame-perfect (Nash) equilibria (SPE or SPNE) in games of complete information. Formal definition: A SPE is a strategy profile where a Nash equilibrium is played in every subgame. For games of perfect information, a subgame is any part of the game tree that starts with a non-terminal node, and includes everything following that node, up to the end of the game tree. See the supplementary material at the end of the next set of slides (about repeated games) for the general definition of "subgame."

ECON 302 (SFU) Backward Induction 9 / 15 Backward Induction

In perfect information games, solving for SPEs is particularly easy: just start at the terminal nodes to infer what players will do at the last step. Given that, figure out what happens at the second-to-last step, and so on. This procedure is called backward induction. When is there a unique SPE in perfect information games? Is every SPE a NE? Is every NE a SPE?

ECON 302 (SFU) Backward Induction 10 / 15 Example

Consider Rock-Paper-Scissors, but suppose player 2 sees what player 1 does before acting. Payoff is 1 for a win, -1 for a loss, and 0 for a tie. Exercise: Draw the game tree. Pure-strategy SPE: (Rock, (Rock Paper, Paper Scissors, Scissors Rock)) (Paper, (Rock→ Paper, Paper→ Scissors, Scissors→ Rock)) (Scissors, (Rock→ Paper, Paper→ Scissors, Scissors→ Rock)) → → → There are infinitely many non-pure SPEs: player 1 is indifferent between all actions, so she can pick any strategy in SPE.

ECON 302 (SFU) Backward Induction 11 / 15 Commitment versus Flexibility

In Battle of the Sexes, players gain from committing to a course of action. As a result, there is a first-mover advantage: Brett would like to threaten that he’dgo to the basketball game if Amy chooses the choir, but cannot do so credibly. As we saw, NE allows for such non-credible threats, while SPE doesn’t. By contrast, in Rock-Paper-Scissors, flexibility creates a second-mover advantage. There are also games where neither is the case.

ECON 302 (SFU) Backward Induction 12 / 15 Example

Player 1 plays T or B. If T, game ends with payoffs (1,0). If B, player 2 plays L or R, leading to payoffs (0,1) and (3,1) respectively. Pure-strategy SPE: (T, L), (B, R) All SPE: (T, (p L, 1 p R)) for any p > 2 − 3 ((q T, 1 q B), ( 2 L, 1 R)) for any q [0, 1] − 3 3 ∈ (B, (p L, 1 p R)) for any p < 2 − 3

ECON 302 (SFU) Backward Induction 13 / 15 Problems with Backward Induction

May not be reasonable when game is long and/or complicated, e.g., . This is a similar problem as in ISD: we assume that players can do long chains of reasoning, that they trust others to do so, that they trust others to trust others to do so, and so on... Even if you accept this assumption, you need a further assumption when a player has multiple best responses at a node: players correctly anticipate what others will do, even though this cannot be deduced by logic alone. This is an assumption we also made for NE. Philosophical aside: unexpected hanging paradox.

ECON 302 (SFU) Backward Induction 14 / 15 Recap

We introduced games of perfect information, and solved them using backward induction. Perfect information: one player acts at a time, and each player sees all previous actions. We represented these games using the extensive form (game tree). Backward induction: start at the bottom of the game tree, figure out the (s) at each node, and work our way up the tree. The resulting strategy profile(s) is/are SPE(s). Next: generalize the concept of to some games without perfect information, so that we can apply SPE there too.

ECON 302 (SFU) Backward Induction 15 / 15