Discrete, Bounded Reasoning in Games
Level-k Thinking and Cognitive Hierarchies
Joe Corliss
Graduate Group in Applied Mathematics
Department of Mathematics
University of California, Davis
June 12, 2015
Joe Corliss (UC Davis)
Discrete, Bounded Reasoning
June 12, 2015
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References
[1] Nagel, Rosemarie. 1995. “Unraveling in Guessing Games: An
Experimental Study.” American Economic Review 85 (5):
1313–1326.
[2] Camerer, Colin F., Teck-Hua Ho, and Juin-Kuan Chong. 2004.
“A Cognitive Hierarchy Model of Games.” Quarterly Journal of
Economics 119 (3): 861–898.
[3] Arad, Ayala and Ariel Rubinstein. 2012. “The 11–20 Money
Request Game: A Level-k Reasoning Study.” American Economic
Review 102 (7): 3561-3573.
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Outline
1. Motivating example
2. Level-k reasoning: theory and experiments
2.1 The Beauty Contest [1]
2.2 11–20 Money Request Game [3]
3. Cognitive hierarchies [2]
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Example: “Guess 2/3 of the Average”
Each player picks a real number in the interval [0, 100]. The winner(s)
are those whose chosen number(s) are closest to 2/3 of the average of
all chosen numbers.
The unique pure NE: all players choose 0. In fact, IEWDA leaves only
the action 0.
Traditional game theory says that all players are perfectly rational, so
we should play 0, right?
Let’s find out.
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Example: “Guess 2/3 of the Average”
What’s the problem? In using IEWDA we assume that all other players
also use IEWDA. Apparently this is a poor assumption in practice.
Indeed, in 2005 the Danish newspaper Politiken played this game with
19,196 of its readers, with a prize of 5,000 Danish kroner. The mean
was 21.6.
Google-translated Politiken headline:
Q: How can we model behavior outside equilibrium? If players don’t
reason “completely,” how much do they reason? Is there a
distribution of “amounts” of reasoning?
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Level-k Reasoning [1], [3]
Level-k reasoning assumes that players form beliefs over opponents’
actions in discrete steps.
Moreover, every player in a game has a “type” describing the amount
of reasoning he does.
A 0-step or level-0 player (L0) forms no beliefs over opponents’
actions. His action depends on the model, and could be:
1. Best-responds to the rules of the game, ignoring opponents; or
2. Plays (uniformly) randomly; or
3. Chooses a salient or intuitive action, a.k.a. a “focal point”
[Schelling, 1960]
For k ≥ 1, a k-step or level-k player (Lk) best-responds to the belief
that all opponents are level-(k − 1).
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Modeling a Game [3]
To develop a level-k model of a game, we need to determine:
1. What an L0 player would do, hopefully unambiguously
2. The distribution of player types
The L0 action recursively determines the actions for each Lk, k ≥ 1.
For (2.), we typically collect some data and choose a distribution of
types to fit the data.
This suggests there is not a universal distribution of types. That is, a
player’s level of reasoning is determined in part by the game.
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The Keynesian Beauty Contest [1]
The general version of “Guess 2/3 of the Average.” Do players exhibit
level-k reasoning?
The game: Each player chooses a real number in the interval [0, 100].
Let 0 ≤ p < 1 be a parameter. The winner(s) are those whose chosen
number(s) are closest to p times the mean of all chosen numbers.
The winner(s) split a fixed prize; all others receive nothing.
Game-theoretic solutions
Unique NE: all players choose 0
IEWDA maximal reduction: {0}
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The Keynesian Beauty Contest [1]
The level-k model:
L0 player: chooses 50. Interpretation: chooses uniformly
randomly over [0, 100], or 50 is a focal point
Lk player, k ≥ 1: chooses 50p k , the best response to L(k − 1)
I
50p k → 0 as k → ∞ (recover game theoretic solution)
Keynes predicted that most players are Lk for some k ≤ 3, and that
higher k is rare [2].
Q: Does player behavior fit into these reasoning “categories”? If so, is
Keynes’ prediction correct?
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The Keynesian Beauty Contest [1]
Experiment: p = 2/3. 66 German students. Small cash prizes.
About 50% fall near step 1 or 2. All higher steps account for less than
10%. Supports Keynes’ idea.
Q: The model fits behavior, but does it explain behavior?
Q: How to account for choices > 50? Mistakes? IEWDA?
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The 11–20 Money Request Game [3]
Arad and Rubinstein design a simple game that naturally triggers
level-k reasoning. Goal: observe unobstructed, unambiguous level-k
reasoning.
The game: “You and another player are playing a game in which
each player requests an amount of money. The amount must be (an
integer) between 11 and 20 shekels. Each player will receive the
amount he requests. A player will receive an additional amount of 20
shekels if he asks for exactly one shekel less than the other player.
What amount of money would you request?” [3]
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The 11–20 Money Request Game [3]
Nice properties of this game for level-k reasoning (compare to Beauty
Contest):
1. Unambiguous level-0 action: 20.
2. Best responses are clear and simple.
3. Best responses are robust to a wide range of beliefs; e.g., 18 is a
best response if > 52.5% of players choose 19.
4. Iterative reasoning is natural; game is defined in terms of best
responses. No other clear decision method*: no pure NE, no
dominated strategies.
*So observed actions are very likely to be the result of iterative
reasoning and not some other decision process.
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The 11–20 Money Request Game [3]
Experiment: The game was played by 108 undergraduate economics
students at Tel Aviv University, who had not studied game theory
before.
After choosing their actions, the subjects were asked to give written
explanations of their choices.
The subjects’ submissions were randomly paired together, and the
payoffs distributed.
Results on the next slide.
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The 11–20 Money Request Game [3]
Observations:
Subjects’ behavior is not near Nash equilibrium. This is
statistically significant (chi-square test, p < 0.0001).
The choice 20 was usually explained as a safety strategy.
Almost all the choices 17–19 were explained in terms of level-k
reasoning.
The choices 11–16 were explained as guesses, rather than 4–9
reasoning steps.
The model with the best statistical fit consists of the player types
L0–L3, and a random/error type (∼ 20%! Perhaps L00 ?)
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The 11–20 Money Request Game [3]
Conclusions
According to the level-k model, subjects did not use more than three
steps of reasoning in choosing their actions.
Past studies of level-k reasoning reached the same conclusion.
However, past games had obstacles to level-k reasoning. This game
was designed to remove all such obstacles.
Q: Why is higher-level reasoning invariably absent in level-k models?
A modification of level-k, called cognitive hierarchies, could explain
this.
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Cognitive Hierarchies [2]
Players are grouped into “types” as before. The level-0 type is
unchanged. Let f (k) be the true distribution of types in a game.
NEW: For k ≥ 1, a level-k player re-normalizes the actual frequencies
to form his beliefs over the other types:
f (q)
,
fk (q) := Pk−1
i=0 f (i)
q ∈ {0, . . . , k − 1}
(“I’m the smartest player!”)
A level-k player, k ≥ 1, best-responds given his beliefs over the other
types.
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Cognitive Hierarchies [2]
Compare:
Level-k
Cognitive Hierarchies
1. Beliefs do not converge.
2. Beliefs are less accurate as k
grows.
3. High-level players act differently.
4. Endless perceived benefits from
thinking. Could reason forever.
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1. Beliefs fk converge to f as
k → ∞.
2. Beliefs are more accurate as k
grows. “Level-∞” is omniscient.
3. High-level players act similarly.
4. Diminishing benefits from
thinking. Reasoning will
terminate.
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Cognitive Hierarchies [2]
How to determine f? If you have no theory, then you have to use
data to estimate the values f (0), . . . , f (k) via maximum likelihood
estimation. But we want a theory.
What properties should f have?
A discrete distribution.
Decreasing term-to-term ratios; in fact, we desire the very
specific property that
f (k)
1
∝
f (k − 1)
k
One distribution very conveniently has exactly this property:
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Cognitive Hierarchies [2]
Poisson. We assume that the types are Poisson distributed with
mean/variance τ > 0, which depends on the player population and
the game.
f (k) :=
e −τ τ k
,
k!
k ≥0
Mode: bτ c if τ ∈
/ Z, or τ and
τ − 1 if τ ∈ Z.
Determine τ by fitting to data.
E.g., “minimize the (absolute)
difference between the predicted
and actual mean of chosen
numbers” [2].
Usually, 1 < τ < 2.
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Cognitive Hierarchies [2], [3]
Revisit the 11–20 game:
In yellow are the
CH-Poisson predictions,
where τ = 2.36 gives a
best fit.
[3] claims all types ≥ 3
choose 17—I think this
is wrong.
Types 3 and 4 choose
17; I think types ≥ 5
choose 16.
Q: Is CH better? Do we
learn more?
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This high-level player
will now answer your questions.
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