lecture 5 - backward induction

Backward InductionEconomics 302 - Microeconomic Theory II: Strategic Behavior

Instructor: Songzi Du

compiled by Shih En LuSimon Fraser University

February 3, 2015

ECON 302 (SFU) Lecture 5 February 3, 2015 1 / 17

Introduction to Sequential Games

Past two weeks: studied games where players move simultaneously.

However, people/firms often make decisions and interact over time.

Example: Battle of the Sexes

If simultaneous move:

GuyBallet Hockey

Girl Ballet 3,1 0,0Hockey 0,0 1,4

What happens if Girl texts Guy: “I’m going to the ballet, and myphone is dying. See you there!”

So if Girl has the opportunity to send that text (and, for whateverreason, Guy doesn’t), will she do it?


The Extensive Form

A good way to represent a sequential game is the extensive form,often called a “game tree.”There is a root node, which represents the beginning of the game.The root is followed by branches. Branches lead to more nodes, whichare followed by more branches. Each branch out of a node is anaction available at that node.Each non-terminal node is a place where the specified player has tomake a decision on the branches/actions.(Sometimes, the Nature makes a probabilistic move at a node; thedifference between Nature and a player is that (1) Nature does nothave any payoff, and (2) the probabilities associated with Nature arefixed.)Each terminal node is an outcome: a combination of actions, justlike before.The numbers below each terminal node are the payoffs from theoutcome corresponding to the node. As usual, the first number isplayer 1’s payoff, the second is player 2’s, etc.ECON 302 (SFU) Lecture 5 February 3, 2015 3 / 17

Perfect Information

Today, we study games of perfect information: one player acts at atime, 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).

After the quiz, we will look at games that do not have perfectinformation.

Example of the latter: playing a prisoner’s dilemma more than once.

Note: don’t confuse perfect information with complete information!


Strategies in Sequential Games

A player’s strategy specifies a probability distribution over her actionsat each node where she plays, regardless of whether that node isreached.

In other words, a strategy is a player’s full contingency plan.

In our example, the Guy’s strategy must include what he would do ifthe Girl’s chooses “Hockey,” even if he doesn’t expect the Girl tochoose “Hockey.”

Just like before, a strategy profile is a collection of each player’sstrategy. So what strategy profile are we predicting?



A strategy is a player’s full contingency plan and must specify whatto do in every contingency (= node).

An extreme example: Bob first decides between jumping off abuilding or not; if not then he decides whether to eat at McDonald orTim Hortons.Jumping off the building alone is an incomplete strategy. Bob’sstrategy must specify “eating at McDonald” or “Tim Hortons”, evenif it specifies “jumping off the building” (which prevents Bob fromeating).

Another example: United States presidential line of succession:1 Vice President of the United States2 Speaker of the House3 President pro tempore of the Senate4 Secretary of State– Acting Secretary of Treasury5 Secretary of Defense6 Attorney General



A strategy is a player’s full contingency plan and must specify whatto do in every contingency (= node).

An extreme example: Bob first decides between jumping off abuilding or not; if not then he decides whether to eat at McDonald orTim Hortons.Jumping off the building alone is an incomplete strategy. Bob’sstrategy must specify “eating at McDonald” or “Tim Hortons”, evenif it specifies “jumping off the building” (which prevents Bob fromeating).Another example: United States presidential line of succession:

1 Vice President of the United States2 Speaker of the House3 President pro tempore of the Senate4 Secretary of State– Acting Secretary of Treasury5 Secretary of Defense6 Attorney GeneralECON 302 (SFU) Lecture 5 February 3, 2015 6 / 17

Nash Equilibrium in Sequential Games

Is our predicted strategy profile a NE?

Are there other pure-strategy NE?

GuyB→B B→B B→H B→HH→B H→H H→B H→H

Girl Ballet 3,1 3,1 0,0 0,0Hockey 0,0 1,4 0,0 1,4

Are these extra pure-strategy NE realistic?


Backward Induction

Idea: should require that players play a best-response (given whatthey 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.

In perfect information games, solving for SPEs is particularly easy:just start at the terminal nodes to infer what players will do at thelast step. Given that, figure out what happens at the second-to-laststep, 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?


Exercise

Consider Rock-Paper-Scissors, but suppose player 2 sees what player1 does before acting.

Payoff is 1 for a win, -1 for a loss, and 0 for a tie.

Draw this game in extensive form, and find its SPE(s) using backwardinduction.


Commitment versus Flexibility

In Battle of the Sexes, players gain from committing to a course ofaction.

As a result, there is a first-mover advantage: the Guy would like tothreaten to go to the hockey game after the Girl has gone to theballet dance, but cannot do so credibly.

As we saw, NE allows for such non-credible threats, while SPEdoesn’t.

Another such example: entry into a market. Incumbent would like todeter entrant by committing to a price war if competitor enters, butthis is often not credible.

By contrast, in Rock-Paper-Scissors, flexibility creates asecond-mover advantage.

There are also games where neither is the case.


Application: Stackelberg Model

Back to oligopoly, quantity competition: suppose there are two firms,and Firm 1 picks quantity before Firm 2. Firm 1 is the industryleader, and Firm 2 is the follower.

For example, Firm 1 commits to quantity by signing a contract withdistributors, or by buying lots of inputs, etc.

Simplest case: both firms have the same constant marginal cost c ,produce a homogeneous good, and face linear market demandP = a− bQ.

We use backward induction to solve for a subgame-perfectequilibrium.


Application: Stackelberg Model (II)

We know from our analysis of the Cournot model that Firm 2’s bestresponse to q1 is

q2(q1) =a− c

2b− q1

2

By backward induction, instead of taking as given a constant q2, Firm1 will take as given Firm 2’s above best response: Firm 1 knows thatq2 now depends on q1.

Firm 1’s profit function:

q1(a− b(q1 + q2)− c)

= q1(a− b(q1 +a− c

2b− q1

2)− c)

=1

2((a− c)q1 − bq21)


Application: Stackelberg Model (III)

Taking the first-order condition and rearranging gives:

q1 =a− c

2b

Plugging back into Firm 2’s best response function gives:

q2 =a− c

4b

Compare to Cournot outcome:

q1 = q2 =a− c

3b


Application: Stackelberg Model (IV)

Stackelberg profits are:

π1 =1

8

(a− c)2

b, π2 =

1

16

(a− c)2

b

Compare to Cournot profits:

π1 = π2 =1

9

(a− c)2

b

Who benefits, and why?

What about the market price?


Centipede Game

Player 1 and 2 are in a partnership: the opportunities are themoney lying on the table. Assume that in the first round, thereare two piles (one for each player): a pile of $4, and a pile of $1.Player 1 has two options in the first round, either to stop (andgrab a pile of money), or to continue the partnership. If he stops,the game ends and he gets $4 while Player 2 gets the remaining$1. If he continues, the game moves to the second round: thetwo piles are doubled (to $8 and $2), and Player 2 face with asimilar decision: stop (the game ends, Player 2 gets $8, andPlayer 1 gets $2), or continue (the piles double again, and Player1 decide at round 3). The game continues for n rounds.

Assume that n = 4. Draw the game tree. Find the SPE by backwardinduction.

How many strategies does Player 1 have? Player 2?


Ultimatum game

There is a dollar to be divided between Alice and Bob. Aliceproposes a division, which is an integer between 0 and 100denoting the amount that she gets in cents (e.g., 50 cents forherself), and Bob gets the rest. After learning Alice’s proposal,Bob responds with an Yes or No. If Bob says yes, Alice’s proposalis implemented; if Bob says no, both of them get nothing.

How many strategies does Alice have? And Bob?

Draw the game tree.

What are the subgame perfect Nash equilibria?


Ultimatum game (answers)

Alice has 101 strategies, and Bob has 2101 strategies.

There are two (pure-strategy) subgame perfect equilibria:

1 Alice proposes 99; Bob accepts Alice’s proposal if and only if it issmaller than or equal to 99.

2 Alice proposes 100; Bob accepts every proposal of Alice.


lecture 5 - backward induction

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