Lecture 6 - Subgame Perfection

Subgame PerfectionEconomics 302 - Microeconomic Theory II: Strategic Behavior

Instructor: Songzi Du

compiled by Shih En LuSimon Fraser University

February 17, 2015

ECON 302 (SFU) Lecture 6 February 17, 2015 1 / 14

Example of Imperfect Information

A pedestrian and a mugger.

The mugger (player 1) moves first, has three actions: bring a gun andshow, bring a gun and not show, not bring a gun.

The pedestrian (player 2) moves second, has two actions: surrender,run.

The pedestrian prefers to surrender if the mugger has a gun, andprefers to run if the mugger does not have a gun. All else equal, themugger prefers no gun to gun.

How do we draw the game tree of such a game?


Mugger-Pedestrian game


Imperfect Information

Sometimes, the player taking an action doesnt know wherehe/she is in the tree, because he/she didnt see how otherplayers played earlier.

Connect these nodes with a dotted line to represent that fact. Such aset of nodes is called an information set.

Information set represents what the player knows up to this point.Thus, the information set is equivalent to a history of plays that isobservable to the player. Each information set also represents acontingency.

Now, a players strategy must specify a course of action foreach information set (rather than each node) where he/sheacts.


Example (contd)

Lets find pure-strategy profiles that make sense as solutions of thisgame.

Common sense that the pedestrian surrenders after the mugger showshis gun.

Then we solve for the NE of the remaining game.


Subgames

Idea: some parts of the game tree can stand alone as a game. Theseare called subgames.

Example: The game we considered: after the mugger shows his gun.

Definition: a node hs successors are all the nodes after h, all theway to the terminal nodes (end of the game tree).

Definition: Suppose you have a game G . A subgame of G consistsof a single non-terminal node and all its successors with theproperty that every information set of G is either entirely insideor entirely outside that set of nodes.

The last part of the definition can be rephrased: no information set ofG contains both nodes inside and nodes outside the set.


Subgames (II)

Way to remember the definition: think of information sets as spiderwebs. Subgames are parts of the tree (except for terminal nodes) thatyou can detach by snapping a single branch and without tearing aweb.

How many subgames were there in our example? How manystrategies?


Subgame-Perfect Equilibrium

Once you understand subgames, the definition of subgame-perfectequilibrium is simple:

A subgame perfect equilibrium is a strategy profile where aNash equilibrium is played in each subgame.

To solve for SPE, do what we have been doing! Start with the smallsubgames toward the end of the tree, and solve bigger and biggersubgames.

Backward induction is a special case of this procedure: in games ofperfect information, every non-terminal node and its successors are asubgame.


Battle of the Sexes with an Outside Option

Recall the battle of the sexes (BoS):Guy

Ballet HockeyGirl Ballet 3,1 0,0

Hockey 0,0 1,3

The girl has an outside option. She can choose out, in which caseshe gets 2, while the guy gets 0. If she chooses in, then the guy(who observes that the girl chose in) and the girl play thesimultaneous-move BoS game.

Analyze this game.


Battle of the Sexes with an Outside Option


Summaries on Battle of the Sexes with an Outside Option

Solve first the smaller subgame, which is a simultaneous-move BoSgame: there are two pure-strategy NEs and one mixed-strategy NE inthe subgame. Then solve the girls choice of In or Out given each ofthese NEs in the subgame.

There are three SPEs:

1 The girl plays (Out, Hockey); the guy plays Hockey.2 The girl plays (In, Ballet); the guy plays Ballet.3 The girl plays (Out, 34Ballet +

14Hockey); the guy plays (

14Ballet +

34Hockey).


Signaling Game: Beer-Quiche

Inspired by the popular 1980s book Real Men Dont Eat Quiche byBruce Feirstein.

Three players: player 1 (strong), player 1 (weak), and player 2.

Player 1, strong or weak, chooses a meal: beer or quiche. Player 2observes this food choice, and chooses to fight or retreat.

Player 2 cannot tell if player 1 is strong or weak, believes that player1 is strong with probability 0.9.

Player 2 gets 1 if he fights the weak player, -1 if he fights the strongplayer, 0 if no fight.

The strong player 1 prefers beer; the weak player 1 prefers quiche.Player 1, strong or weak, gets 2 if Player 2 doesnt fight him, 0otherwise; player 1, strong or weak, gets an additional 1 if he eats hispreferred meal.

Draw the game tree, identify the subgame(s) and the strategies, andfind the SPEs.


Beer-Quiche game


Summaries on Beer-Quiche game

To find a SPE, start with a conjecture on the strategy of Player 1,and calculate Player 2s best responses. For each of these bestresponses, verify that both types of Player 1 are best responding inthe conjectured strategy.

There are two pure-strategy SPEs:

1 (S-Beer, W-Beer, B-Retreat Q-Fight)

2 (S-Quiche, W-Quiche, B-Fight Q-Retreat)

Intuition: the weak player 1 must camouflage himself with thestrong player 1 to force player 2 to retreat. This is called a poolingequilibrium.


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Lecture 6 - Subgame Perfection