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Game theory�

Game theory: A mathematical approach to modeling behavior by analyzing the strategic decisions made by interacting rational players

Standard game theory (Nash 1950) assumes:

Players are homo economicus

Recall, homo economicus:

Perfectly rational
Maximizes expected utility
Cares only about monetary incentives
Selfish (self-regarding preferences)

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Iterated elimination equilibrium�

Iterated elimination equilibrium (IEDS): Equilibrium obtained by eliminating dominated strategies

Often it is called the IEDS equilibrium (iterated elimination of dominated strategies)

Solving games using dominated strategy method:

If a strategy is dominated, eliminate it
The size and complexity of the game is reduced
Then eliminate any dominated strategies from the reduced game
Continue doing so

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Subgame perfect equilibrium�

Subgame perfect equilibrium (SPNE): Equilibrium obtained by backward induction

Often it is called the SPNE equilibrium (subgame perfect Nash equilibrium)

Solving games using backward induction:

Start from the very end
Find the best response (i.e., the strategy that gives the highest payoff)
Refine the game, taking into account the results from previous steps
Continue until you cannot refine the game

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Limitations of IEDS and SPNE�

People are not perfectly rational:

Make mistakes
Not good at backward induction
Need time to learn

People are not perfectly selfish:

Other-regarding preferences

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Behavioral game theory�

Behavioral game theory: Extends standard game theory by taking into account that people have limited strategic abilities and they care about payoffs of others

Behavioral game theory (Camerer 2003) assumes:

Players are bounded rational
Players care about payoffs of others
Players have difficulty learning

Typical games:

Ultimatum game
Dictator game
Trust game
Prisoners’ Dilemma
Cooperation game

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Level-k Model

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Level-k thinking�

Level-k thinking: Decision-making process which is based on the assumption that players have different degrees of rationality

It is a non-equilibrium concept that describes how people actually behave

Solving games using level-k thinking:

Level 0: Non-strategic (random, reference point, etc.)
Level 1: Best respond to level 0
Level 2: Best respond to level 1
Level 3: Best respond to level 2

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Level-k thinking�

Princess Bride:

“Battle of wits”
https://www.youtube.com/watch?v=U_eZmEiyTo0

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Example: Guessing game�

Each person in a group is asked to choose a number between (and including) 0 and 100 simultaneously. Communication is not allowed. The person whose number is closest to, but not exceeding, 2/3 of the average (called the target number) earns $10, while the rest earns $0.

The game-theoretic (Nash) equilibrium of 0 obtained from iterated elimination of dominated strategies is a poor predictor about individual choices

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Example: Guessing game�

Solving the game using level-k thinking (Nagel 1995)

First, determine the behavior of Level 0:

Level 0: random answer = 50 (average 0-100)

Second, determine the behavior of Level k as the best response to Level k-1:

Level 1: answer = 33 target = 50 × 2/3 = 33
Level 2: answer = 22 target = 33 × 2/3 = 22
Level 3: answer = 15 target = 22 × 2/3 = 15
Level 4: …
(Nash) equilibrium: answer = 0 target = 0 × 2/3 = 0

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Experiment #10 results�

Class experiment:

Guessing game

Nash equilibrium

Level 0

Level 1

Level 2

Level 3

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Example: Centipede game�

Solving the game using backward induction:

The SPNE equilibrium is (S, S’) with payoffs of (2, 0): player 1 stops (S) at the first round and player 2 stops (S’) at the second round

What if the game is longer?

The SPNE equilibrium is (SS’’S’’’’, S’S’’’S’’’’’) with payoffs of (2, 0): player 1 stops in each round and player 2 stops in each round

S’

C’

2, 0

1, 3

4, 2

S’

C’’’

C’’

C’

2, 0

3, 5

1, 3

4, 2

C’’’’’

C’’’’

5, 7

8, 6

6, 4

S’’’

S’’

S’’’’’

S’’’’

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Example: Centipede game�

Solving the game using level-k thinking (Kawagoe and Takizawa 2012)

Behavior of Level 0:

Random play would imply choosing C or S with probability 0.5

Behavior of Level 1 as the best response to Level 0:

Player 1 chooses CC’’C’’’’ as the best response to Level 0 Player 2
Player 2 C’C’’’S’’’’’ as the best response to Level 0 Player 1

S’

C’’’

C’’

C’

2, 0

3, 5

1, 3

4, 2

C’’’’’

C’’’’

5, 7

8, 6

6, 4

S’’’

S’’

S’’’’’

S’’’’

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Example: Centipede game�

Behavior of Level 2 as the best response to Level 1:

Player 1 chooses CC’’S’’’’ as the best response to Level 1 Player 2
Player 2 C’C’’’S’’’’’ as the best response to Level 1 Player 1

Behavior of Level 3 as the best response to Level 2:

Player 1 chooses CC’’S’’’’ as the best response to Level 2 Player 2
Player 2 C’S’’’S’’’’’ as the best response to Level 2 Player 1

S’

C’’’

C’’

C’

2, 0

3, 5

1, 3

4, 2

C’’’’’

C’’’’

5, 7

8, 6

6, 4

S’’’

S’’

S’’’’’

S’’’’

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Example: Centipede game�

Behavior of Level 4 as the best response to Level 3:

Player 1 chooses CS’’S’’’’ as the best response to Level 3 Player 2
Player 2 C’S’’’S’’’’’ as the best response to Level 3 Player 1

Behavior of Level 5 as the best response to Level 4:

Player 1 chooses CS’’S’’’’ as the best response to Level 4 Player 2
Player 2 S’S’’’S’’’’’ as the best response to Level 4 Player 1

S’

C’’’

C’’

C’

2, 0

3, 5

1, 3

4, 2

C’’’’’

C’’’’

5, 7

8, 6

6, 4

S’’’

S’’

S’’’’’

S’’’’

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Example: Centipede game�

Behavior of Level 6 as the best response to Level 5:

Player 1 chooses SS’’S’’’’ as the best response to Level 5 Player 2
Player 2 S’S’’’S’’’’’ as the best response to Level 5 Player 1

Behavior of higher levels is going to be the same as the behavior of Level 6, which also corresponds to the SPNE

Recall, the SPNE equilibrium is (SS’’S’’’’, S’S’’’S’’’’’)

S’

C’’’

C’’

C’

2, 0

3, 5

1, 3

4, 2

C’’’’’

C’’’’

5, 7

8, 6

6, 4

S’’’

S’’

S’’’’’

S’’’’

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Experiment #11 and #12 results�

Class experiment:

Centipede game

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Evidence of level-k thinking�

Typical distribution of levels in experiments (Crawford et al. 2013):

Level 0: about 10%
Level 1: about 40%
Level 2: about 30%
Level 3: about 10%
Other: about 10%

People who are trained to behave rationally, behave as high level k (Palacios-Huerta and Volij 2009; Levitt et al. 2011):

70% of chess players and 100% of chess Grandmasters choose to stop

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Limitations of level-k models�

Hard to define Level 0

Random
Reference point

Too many parameters (degrees of freedom)

The percentage of each level is an additional degree of freedom
Needs more (ad hoc) assumptions in order to constrain degrees of freedom

Low predictive power

Are types fixed across games?
Georganas et al. (2015) estimate subjects’ types in two different games and find no correlation in estimated types

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References�

Dhami, S. (2016). The Foundations of Behavioral Economic Analysis. Oxford University Press.
Nagel, R. (1995). Unraveling in guessing games: An experimental study. American Economic Review, 85, 1313-1326.
Kawagoe, T., & Takizawa, H. (2012). Level-k analysis of experimental centipede games. Journal of Economic Behavior & Organization, 82, 548-566.
Crawford, V.P., Costa-Gomes, M.A., & Iriberri, N. (2013). Structural models of nonequilibrium strategic thinking: Theory, evidence, and applications. Journal of Economic Literature, 51, 5-62.
Levitt, S.D., & List, J.A., & Sadoff, S. (2011). Checkmate: Exploring Backward. Induction among Chess Players. American Economic Review, 101, 975-990.
Palacios-Huerta, I., & Volij, O. (2009). Field centipedes. American Economic Review, 99, 1619-1635.
Georganas, S., Healy, P. J., & Weber, R. A. (2015). On the persistence of strategic sophistication. Journal of Economic Theory, 159, 369-400.