Transcript Nagel: Unraveling in Guessing Games: An Experimental Study Economics 328 Spring 2005 What is Game Theory?  Game theory is the study of strategic interaction among.

Nagel:
Unraveling in Guessing Games: An
Experimental Study
Economics 328
Spring 2005
What is Game Theory?
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Game theory is the study of strategic interaction among individuals or
groups.
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I have to anticipate what others will do in order to determine my own best
strategy.
I should anticipate that others will be trying to figure out what I’m going to do.
Examples of Games
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Many industries are dominated by a small number of large firms. In deciding on a
competitive strategy, such firms cannot take the market as fixed. Instead, firms
should anticipate other firms responding to their actions.
Rather than being driven by ideology, many politicians are quite strategic in
choosing their positions. In choosing a position, politicians must anticipate not only
how the electorate will react but also how opposing politicians will react.
Many situations from everyday life have strong strategic elements. For example,
think about choosing where to go with your S.O. on the weekend. What are
strategic elements of this situation?
What is Game Theory?
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Game theorists try to develop simple models of
strategic situations and then predict how individuals
will behave in these games. These models almost
always rely strongly on the rationality of players.
Experiments have played two important roles in the
development of game theory.
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Experiments have unequivocally demonstrated that most
individuals are not as rational as game theorists might like.
Experiments have helped identify systematic departures
from rationality. By building these systematic departures
into the theory, we can hopefully build a theory with greater
predictive power.
Normal Form Games
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Definition: A normal form game consists of the
following elements:
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A set N of players. Let n be the number of players in N.
For each player i, a set Si of available strategies. A strategy
summarizes every action a player will take under any
conceivable situation in the game.
For each player i, a payoff function ui(s1,s2,…,sn). This
function gives player i's payoffs as a function of his/her
strategy and the strategy choices of all n-1 other players.
Often times, we represent normal form games in
tabular form.
Normal Form Games -- Example
Consider an arbitration case between a firm and an employee (represented by their union).
Each side presents a case and a proposed settlement. The arbitrator then must choose one of
the two proposed settlements. Both the firm and the union have the option of using lawyers
for the case. The choice of whether or not to use a lawyer must be made simultaneously –
neither side knows the other’s choice prior to making a decision. Suppose each side cares
about the likelihood of a “win,” with W being the value of a win, and the cost of hiring a
lawyer, cL. Empirical work by Ashenfelter and Bloom indicates that the resulting game is
given by the table below. Employer payoffs are listed first.
Union
Employer
Lawyer
No Lawyer
Lawyer
.46W – cL, .54W - cL
.23W, .77W - cL
No Lawyer
.73W – cL, .27W
.44W, .56W
Dominance
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One of the simplest solution concepts for games is dominance.
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Individuals should never use a strategy if the exists another option
that always does (weakly) better.
This requires only a low level of rationality.
Definition: A strategy si is (weakly) dominated for player i if he
has another strategy si’ such that si never gives a (weakly) greater
payoff than si’ regardless of what strategies are chosen by the
other players.
Example: In the arbitration game, suppose .02W  cL  .21W.
For both sides it is a dominant strategy to use a lawyer. This is
an example of a prisoners’ dilemma – both sides would be
better off if they could coordinate on not using lawyers.
Iterated Removal of Dominated Strategies
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The set of serially undominated strategies is defined iteratively.
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Start with some normal from game {N,S,u}.
Form a new game by removing all of the weakly dominated strategies from the old
game.
Take the new game and form a third game by removing all of the weakly dominated
strategies from the new game.
This process continues until we can’t remove any more strategies. The remaining
strategies are called the set of serially undominated strategies.
The process of removing strategies is called iterated removal of weakly
dominated strategies.
Like dominance, iterated removal of dominated strategies requires rationality.
Moreover, common knowledge of rationality is also required.
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I am rational.
I know you are rational.
I know you know I am rational
I know you know I know you are rational.
And so forth, ad infinitum
The Beauty Contest Game
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Consider Nagel’s guessing game, also
known as the beauty contest game.
The strategy set available to each
player in this game is the numbers
between 0 and 100 (inclusive). The
player closest to (for example) ½ the
median number chosen wins a prize of
$1. All other individuals win nothing.
In case of a tie, the prize is split
evenly. We can show that zero is the
only serially undominated strategy.
Why is it the “beauty contest” game?
The Beauty Contest Game
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Round 1: The median can never be greater
than 1000. Therefore, picking 500 weakly
dominates any higher number. Remove the
numbers greater than 500.
Round 2: Given that only the numbers from
0 to 500 remain, the median can never be
greater than 500. Therefore, picking 250
weakly dominates any higher number.
Remove the numbers greater than 250.
Round 3: Given that only the numbers from
0 to 250 remain, the median can never be
greater than 250. Therefore, picking 125
weakly dominates any higher number.
Remove the numbers greater than 125.
We can continue this process of elimination
ad infinitum. The only number that won't
eventually be eliminated is zero.
Nagel: Unraveling in Guessing Games
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Research Questions: The guessing game is good setting for seeing how
subjects reason about games. The main question is whether subjects will
behave as rationally as the theory predicts. If not, can we see subjects learning
over time? In particular, will they get better at anticipating the actions of
others?
Initial Hypotheses: Rather than being fully rational, Nagel expected subjects to
only go a few steps into the chain of reasoning. More technically, she expected
to see play in the first game clustered around 100*pn, where n is some integer,
and then clustered around R*pn in later rounds, where R is a reference point.
She expected subjects to learn over time, possibly even going more steps into
the chain of reasoning.
Experimental Design: Nagel ran the guessing game with subjects guessing
between 0 and 100. She ran sessions with p = 1/2 (3 sessions), p = 2/3 (4
sessions, and p = 4/3 (3 sessions). Note that iterated dominance no longer
applies with p = 4/3. Sessions contained 15 – 18 subjects. Each session played
the game four times with standard feedback. Sessions were fairly short (about
45 minutes) and the prize was substantial.
Experimental Results
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Substantial clustering is
observed, as predicted by a
steps of reasoning approach. In
the first game, most of the
clustering takes place at steps
0, 1, and 2. Choosing step 2 is
in fact typically optimal given
the choices of the population –
playing the equilibrium would
actually lead to zero earning.
Notice that with p = 4/3,
subjects go fewer steps into the
reasoning – one step is the
modal choice rather than two
steps (as for p = ½ and p =
2/3).
Experimental Results
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Subjects clearly move towards lower choices over time in the p
= 1/2 and p = 2/3 treatments. They move toward higher
choices over time in the p = 4/3 treatment.
Experimental Results
Experimental Results
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Clustering is not as clear in later round, possibly because it is less easy to identify the
reference point. There is little evidence of increased depth of reasoning with experience.
Nagel claims that her data support the directional learning hypothesis. My opinion is that
this hypothesis is poorly developed and that her results are indistinguishable from
regression to the mean.
Follow-up Papers
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Camerer, Ho, and Weigelt (1998) revisit the guessing game (renamed the pbeauty contest game). They add several details to Nagel’s findings:
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They use a more rigorous method than Nagel to estimate levels of reasoning, and find
somewhat lower levels of reasoning (more level 0s, fewer level 2s and 3s).
They study the ability of subjects who are experienced with one p-beauty contest
game to play a different p-beauty contest game. They find some evidence of positive
transfer – initial behavior is the same for experienced and inexperienced subjects, but
experienced subjects learn faster.
They study the learning process more carefully than Nagel. They find evidence that a
large fraction of the population (about 70%) is able to anticipate learning by others.
Bosch-Domènech, Montalvo, Nagel, and Satorra (2002) report results from a
series of newspaper experiments with the guessing game (p = 2/3).
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In spite of the lack of control, the newspaper experiments look remarkably similar to
the laboratory experiments with spikes at 33, 22, and 0.
Subjects who were able to correctly describe the Nash equilibrium generally (81%) did
not choose zero.
Subjects who conducted their own experiments did better than game theorists or the
general public at predicting the winning number!