Transcript Introduction - Editorial Express

Correlated equilibria, good and
bad: an experimental study
John Duffy (University of Pittsburgh)
Nick Feltovich (University of Aberdeen)
[email protected]
The preliminary paper can be found at
www.abdn.ac.uk/~pec214/papers/correl.pdf
Background
Player 2
Consider this version of the
Chicken game:
Player 1
Nash equilibria are (D,C),
(C,D), and a mixed
equilibrium with
Prob(C)=3/5.
D
C
D
0,0
9,3
C
3,9
7,7
The mixed NE is symmetric, but yields low payoffs (5.4 for
each player).
Are there equitable outcomes where the players do better?
Background
Player 2
Aumann (1974): If the players
can observe a public coin
D
C
toss, they can get payoffs of 6
Player 1 D
0,0
9,3
each, instead of the mixed
NE payoff of 5.4 each.
C
3,9
7,7
E.g. Player 1 chooses C after
Heads, D after Tails; Player 2
chooses D after Heads, C
after Tails.
In this case (unlike mixed-strategy Nash equilibrium), players’
strategies are not statistically independent.
Correlated equilibrium: generalisation of Nash equilibrium that
allows strategies to be correlated across the players.
Background
Player 2
D
C
0,0
9,3
Aumann (1974): If there is a
third party (non-strategic Player 1 D
player), the players can do
C
3,9
7,7
better still.
Suppose the third party chooses the outcomes (D,C), (C,D),
and (C,C) with equal probability, and “recommends” the
corresponding action to each player (but not the
opponent)—and suppose the third party’s behaviour is
common knowledge between Players 1 and 2.
Then, if the players follow their recommendations, expected
payoffs are 6-1/3 each.
Background
Not only does following these
recommendations raise
players’ payoffs, it is
equilibrium behaviour!
Why?
Player 2
D
C
Player 1 D
0,0
9,3
C
3,9
7,7
Suppose Player 1 receives a D recommendation. Then she
knows for sure that Player 2’s recommendation is C. If she
expects Player 2 to choose C, then U1(D)=9>7=U1(C).
Suppose Player 1 receives a C recommendation. Then she
knows that Player 2 got either a C or a D recommendation
with probability 0.5. If she expects Player 2 to follow his
1
1
recommendation, then U1 (C)  (3)  (7)  5 and
2
2
1
1
U1 (D)  (0)  (9)  4.5, so U1(C)>U1(D).
2
2
Background
Correlated equilibrium allows for higher symmetric payoffs than
in any Nash equilibrium—but also for lower symmetric payoffs.
9
6
Player 2
payoff
3
0
0
3
6
Player 1 payoff
9
Research questions
(a) Is it possible to implement correlated equilibria when actual
people are playing this game? (I.e., do people follow
recommendations?)
(b) Does the answer to (a) depend on which outcome
distribution is chosen?
We consider two correlated non-Nash equilibria:
“Good recommendations”:
“Bad recommendations”:
Player 1 D
C
Player 2
D
C
0
1/3
1/3
1/3
Expected payoff: ≈6.333
Player 2
D
C
Player 1 D
1/5
2/5
C
2/5
0
Expected payoff: 4.8
Research questions
We also consider:
“Nash recommendations”
(correlated equilibrium that
is a convex combination of
Nash equilibria):
Player 1 D
C
Player 2
D
C
0
1/2
1/2
Expected payoff: 6
0
“Very good recommendations”
(not a correlated equilibrium):
Player 2
Player 1 D
C
D
C
0
1/10
1/10
4/5
Expected payoff: 6.8
Previous research
Cason and Sharma (2007 Economic Theory): Similar game,
correlated equilibrium with good recommendations.
Player 2
Left
Right
Player 1 Up
3,3 (0.000)
48,9 (0.375)
Down 9,48 (0.375) 39,39 (0.250)
Results:
 Subjects followed Up/Left (D) recommendations roughly
85% of the time, Down/Right (C) recommendations
roughly 75% of the time.
 Adding recommendations to the game raised average
payoffs by about 15% (vs. predicted 35%).
Experimental design
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Experimental sessions took place at Pittsburgh Experimental
Economics Laboratory (PEEL), with all interaction via
networked computers.
Subjects were randomly matched in each round (12 subjects in
each session).
All subjects played 20 rounds without recommendations, 20
rounds with recommendations (within-subject variation).
Order of games (with recommendations or without)—varied
across subjects.
Good, bad, Nash, or very good recommendations—varied
across subjects.
Experimental design

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
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Instructions—including the outcome distribution underlying
the recommendations—were presented orally (as well as in
writing), in an attempt to satisfy common knowledge of the
situation.
Actions labelled as “X” (=D) and “Y” (=C).
Feedback: own and opponent recommendation, action, payoff
at the end of each round
Payments: $5 show-up fee plus $1/point earned in two
randomly chosen rounds (one from rounds 1-20, one from
rounds 21-40).
Average payment of roughly $15 for a 45-60 minute session.
Recommendations explained in neutral language.
Experimental design
Excerpt from our instructions (good-recommendations treatment):
“Recommendations: Before choosing an action in a round, both you and the person
you’re matched with are given recommendations by the computer program. Different
recommendations will be given in different rounds. In any round, there are three
possibilities:
• There is a ⅓ (33⅓%) chance that it will be recommended that you choose [D] and the
other player choose [C];
• There is a ⅓ (33⅓%) chance that it will be recommended that you choose [C] and the
other player choose [D];
• There is a ⅓ (33⅓%) chance that it will be recommended that both you and the other
player choose [C];
It will never happen that you are both recommended to choose [D].
These recommendations are optional; it is up to you whether or not to follow them.
[Emphasis added.]
Notice that your recommendation may give you information about the recommendation
that was given to the person matched to you.”
Comparison with other procedures
Excerpt from Cason/Sharma instructions:
“Why you should follow the recommendations
You should follow the recommendation given by the computer, because as long as the
person you are paired with also follows his or her recommendation then you earn
more on average by following the recommendation. Here is why:
1. First, remember that if both you and the participant you are paired with follow the
recommendations, you will never have the worst Up-Left outcome (in which both
participants earn only 3), because that outcome is never recommended.
2. Next, if you are a Red participant and you receive the recommendation to choose Up,
then you know that the Blue participant you are paired with has received the
recommendation to choose Right, since the outcome Up-Left is never recommended.
[You know that a green or red ball was not drawn, since they recommend Down.] If
this Blue participant follows his recommendation and chooses Right, then you earn
more by following your recommendation to choose Up (48) than by not following
your recommendation and choosing Down (39).
[…]
To reiterate: you always earn more by following your recommendation as long as the
participant you are paired with also follows his or her recommendation.”
Experimental results
Aggregate outcome frequencies [implied from recommendations]:
(C,C)
(C,D) or (D,C)
(D,D)
Mean payoff
.347
.494
.159
5.393
Good
recommendations
.281
[.333]
.579
[.667]
.140
[.000]
5.444
[6.333]
Bad
recommendations
.327
[.000]
.481
[.800]
.192
[.200]
4.902
[4.800]
Nash
recommendations
.323
[.000]
.565
[1.000]
.112
[.000]
5.648
[6.000]
Very good
recommendations
.325
[.800]
.466
[.200]
.208
[.000]
5.075
[6.800]
.360
.480
.160
5.400
No recommendations
Mixed NE prediction
Experimental results
Frequency of followed
Frequency of followed
Experimental results
Parametric statistics—methodology
 Probit models
 Dependent variable: indicator for C choice (1=yes, 0=no)
 Independent variables:
— round number; indicator for recommendations-first ordering
— indicator for C recommendation, product with round number
— indicator for D recommendation, product with round number
 Individual-subject random effects, some models with session fixed effects
 Separate estimations for each treatment
 STATA (v. 10)
We estimate the incremental effect of a C recommendation on the
likelihood of a C choice in round t:
Φ(X∙B + βCrec + βCrec∙round ∙ t) − Φ(X∙B)
(analogous formula for D recommendation)
Experimental results
Estimated incremental effect of recommendations on C choice
Experimental results
Are individual subjects better off if they follow recommendations?
To answer this, we consider foregone payoffs:
(Payoff from choosing other action) − (Payoff actually earned)
=> Do subjects who follow recommendations more often have lower
foregone payoffs?
Experimental results
Association between following recommendations and foregone payoffs
(individual subjects, 20-round averages):
Good, Nash recommendations
Bad, very good
recommendations
Summary
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Giving recommendations to subjects has an effect on the
distribution of outcomes.
Recommendations are followed more often than chance
(mixed-strategy Nash equilibrium) would predict, but far less
than 100%.
The frequency of followed recommendations varies by
treatment—they are more likely to be followed when
1.
they come from a correlated equilibrium;
2.
they raise payoffs.
Next steps

Next paper: “endogenous correlated equilibrium”—will a strategic
third party make recommendations that form a correlated
equilibrium? If so, what kind of correlated equilibrium?