Introduction to Game

Simultaneous
Simultaneous games
Simultaneous – actions taken at exactly the same time or in isolation – imperfect information (imperfect knowledge).
Examples:
Pricing
Designing new product
Elections
Football (goal keeper (goalie) and striker during penalty kick)

1. Depicting simultaneous-move games with discrete strategies
Simultaneous games
Strategy = action (in pure strategy games)
Game table, game matrix, payoff table, payoff matrix
The matrix is called the normal form (strategic form) of the game

Simultaneous games
1. Depicting simultaneous-move games with discrete strategies
Let us have a look at {L, M}

Simultaneous games
3. Dominance
Prisoner’s dilemma

3. Dominance
Simultaneous games
Prisoner’s dilemma
Dominant vs. dominated strategy
Dominant strategy – one that outperform all other strategies available
Dominant strategy – my action is done regardless of my belief of the other player’s choice

A. Both players have dominant strategies
Simultaneous games
Prisoner’s dilemma (PD)
Three essential features:
- each player has two strategies: cooperate and defect from cooperation
- each player has dominant strategy
- dominance solution equilibrium is worse for both players

Simultaneous games
B. Only one player has a dominant strategy
Analysis of pick a color games
2a. Pick a color

Simultaneous games
B. Only one player has a dominant strategy
Analysis of pick a color games
2b. Pick a color

Simultaneous games
C. Successive elimination of dominated strategies
Dominance solvable – solution achieved via successive (iterated) elimination of dominated strategies

Simultaneous games
C. Successive elimination of dominated strategies
Weak vs. strict dominance
Using weak dominance we may skip some NE – cell-by-cell check is necessary before elimination
Find NE
How many NE?
One?
NO!
Three

Simultaneous games
4. Best-response analysis
If best-response analysis of discrete strategies does not lead to NE then, there are no NE in pure strategies.

Simultaneous games
5. The minimax method for zero-sum games
Logic of strict conflict (zero-sum)
What is good for me is bad for the other player.
max=10
max=5,6
max=13
min=2
min=5,6
min=1
min=-2

Finding NE in pure strategies:
Simultaneous games
Cell-by-cell inspection
Dominant strategies
Successive elimination of dominated strategies
Minimax (zero sum games)
Best-response analysis

6. Three players
Simultaneous games
Emily’s preferences: (others are the same)
6 – don’t contribute, Talia & Nina contribute
5 – all contribute
4 – don’t contribute, one of the others contributes
3 – contribute, one of the others contributes
2 – don’t contribute, Talia & Nina do not
1 - contribute, Talia & Nina do not

Simultaneous games
6. Three players
TALIA chooses
Contribute
Don’t Contibute

6. Three players
Simultaneous games
Searching for NE
Check for dominant strategies (check in two pages!)
Cell-by-cell inspection NE
Best response NE

Simultaneous games
6. Three players- best response
TALIA chooses
Contribute
Don’t Contribute

Simultaneous games
7.Multiple equilibria in pure strategies
Pure coordination. „flat tire”
Another example:

7.Multiple equilibria in pure strategies
Simultaneous games
Pure coordination. Players want to coordinate on the same action, no matter what action.
Coordination device - focal point
Expectations of the players must converge on focal point.
One must knows, that the other knows, that the first knows, that the other …

Simultaneous games
7.Multiple equilibria in pure strategies
Assurance

7.Multiple equilibria in pure strategies
Simultaneous games
Assurance – multiple equilibria, but players prefer the same equilibrium
Focal point – the obvious equilibrium to choose.
Focal point requires convergence of expectations, otherwise they may fail to coordinate. But assurance (that the other player chooses proper action) is relatively easy to obtain.

Simultaneous games
7.Multiple equilibria in pure strategies
The Battle of Sexes

7.Multiple equilibria in pure strategies
Simultaneous games
The Battle of Sexes – both equilibria are preferred over the other possible outcome of the game, but
each player prefers different equilibrium.
If both players are nice – they end up in bad place.
Strategy is needed.
In repeated games – coordination may be negotiated and maintained.

Simultaneous games
7.Multiple equilibria in pure strategies
Chicken game

7.Multiple equilibria in pure strategies
Simultaneous games
Chicken game – multiple equilibria, but each player prefers different equilibrium and they want to avoid choosing the same action.
Features:
two strategies: “tough” and “weak”
two pure strategy NE
each player prefers strictly, the other plays “weak”
payoff for “tough” strategies played by both is very bad for them