# Essay about 20005 Lecture2 SeqGames

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ECON 20005 / 316-210
Competition and Strategy
Topic 2: Sequential Games
David Byrne
Department of Economics
University of Melbourne

Recommended reading in DS: 2nd edition, chapter 3 (p. 45–72,
77–78), or 3rd edition, chapter 3 (p. 47–72, 79–80).
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Sequential Games

Sequential move games are played by two or more players for two or more periods

Ultimatum game is a sequential game

Chess is a sequential game

Paper-Rock-Scissors is not a sequential game

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Representing Sequential Games: Game Trees

Sequential move games are normally presented as game trees.

Game trees are often referred to as extensive form games.

Think of these as a possible paths of players’ actions and outcomes ◮

Game trees reveal the players, their actions, the timing of their actions and the payoffs.

Game trees vs. decision trees: Game trees are joint decision trees for all the players in a game.

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Sir Richard Branson

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Game Trees
Example

Consider the phone company choice game represented by the following game tree with two players, Husband (H) and Wife (W):
H

Telstra
4, 4

Telstra
W
Virgin
3, 5

Virgin
W
Telstra
4, 3

Virgin
6, 6

So if the actions are H:{Telstra} and W:{Virgin}, H gets 3 and W gets 5. That is, wife prefers Virgin if husband prefers Telstra.

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Game Trees
Game characteristics

This is a non-zero sum game.
If actions are H:{Telstra} and W:{Virgin}, then H gets 3 and
W gets 5
If actions are H:{Virgin} and W:{Virgin}, then H gets 6 and
W gets 6
So we see pay-offs from actions do not result in one player’s gain = other player’s loss. Here, both do better if H:{Virgin} and W:{Virgin} then if H:{Telstra} and W:{Virgin}!

This is a complete information game.
W sees all possible previous moves by all other players (H)
H can anticipate any reaction of W to H’s choice

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Game Trees
Characterization of a Game Tree

Players: Two players – Husband (H) and Wife (W)

Nodes: Three decision nodes and four terminal nodes

Timing: Sequential, player H moves first, then player W

Outcomes and Payoffs: At each terminal node, payoffs for all players are listed for that sequence of moves. They are normally listed in the order of who moves first.

Actions are moves taken at decision nodes, where each branch represents a possible action.

Strategies: action plans that describe a player’s actions at all of his/her decision nodes

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2 Things that Drive Your Lecturer Nuts
STRATEGIES ARE ACTION PLANS!

H

Telstra
4, 4

Telstra
W
Virgin
3, 5

Virgin
W
Telstra
4, 3

Virgin
6, 6

Example strategy: H Virgin) and W is (Virgin, Telstra)

They are NOT payoffs (i.e, (4,3) from the example strategy)

They are NOT the sequence of decisions implied by action plans (i.e., (Virgin, Telstra) from the example strategy)

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Game Trees
Backward Induction

A subgame is a game comprising a portion of a larger game, starting at a non-initial node of the larger game.

Backward induction asks to start at the final subgames, and to work backwards towards the initial node.
(That’s why backward induction is also known as rollback.)

Rational player selects in every subgame the move that maximises their own payoff.

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Game Trees
Example

The phone company choice game has three subgames:

The two subgames for each of the possible (W) choices

The one subgame for the one (H) choice (i.e., the entire game!) H

Telstra
4, 4

Telstra
W
Virgin
3, 5

Virgin
W
Telstra
4, 3

Virgin
6, 6

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Game Trees
Backward Induction

To start solving the phone company game, consider the two final subgames:
When player H has chosen Telstra, player W gets payoff 4 if she chooses Telstra and 5 if she chooses Virgin.
When Player H has chosen Virgin, player W gets payoff 3 if she chooses Telstra and 6 if she chooses Virgin.

H

Telstra
4, 4

Telstra
W
Virgin
3, 5

Virgin
W
Telstra
4, 3

Virgin
6, 6

Hence, the best strategy of player W is (Virgin, Virgin)
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