COGNITION AND BEHAVIOR IN TWO-PERSON
GUESSING GAMES: AN EXPERIMENTAL STUDY
Miguel A. Costa-Gomes
University of York
Vincent P. Crawford
University of California, San Diego
April, 2005
OVERVIEW
I) INTRODUCTION
II) EXPERIMENTAL DESIGN
III) ECONOMETRIC MODEL
IV) SOME FINDINGS
INTRODUCTION
ACTIONS
INFORMATION SEARCH
ACTIONS & INFORMATION SEARCH
INTRODUCTION
PREVIOUS STUDIES OF GUESSING GAMES
Nagel (1995, AER), Ho, Camerer, and Weigelt (1998, AER), and Grosskopf
and Nagel (2001).
In their guessing games, n players make simultaneous guesses between lower
and upper limits [a, b]. The player whose guess is closest to a target (p) times
the group average wins a prize. The structure is public knowledge. The same
game is repeated a number of times.
a
Nagel
b
n  2 (15 to 18), p  1/ 2, 2 / 3, or 4 / 3, and limits [0,100].
Ho, Camerer, Weigelt n  2 (3 to 7), p  0.7, 0.9, 1.1, or 1.3, and [0,100] or [100,200].
Grosskopf and Nagel
n  2, p  2 / 3, and limits [0,100].
EXPERIMENTAL DESIGN
TWO-PERSON GUESSING GAMES
Each player has a lower and an upper limit (both strictly positive), but players
are not required to guess between their limits. Guesses outside a player’s limits
are automatically adjusted up to his lower limit or down to his upper limit as
necessary, R(a i ,bi ; x i )  min{bi ,max{a i ,xi }}.
ai
aj
bi
bj
Each player also has a target p,i and his payoff increases with the closeness of
his adjusted guess to his target times the other’s adjusted guess.
ei | R(a i , bi ; x i ) - p i R(a j , b j ; x j ) |
EXPERIMENTAL DESIGN
TWO-PERSON GUESSING GAMES
Example – Game g:
p i  1.5
300
900
p j  0.5
100
900
Suppose i guesses 400 --> j guesses 200 --> i guesses 300 --> j guesses 150
--> i guesses 225 (adjusted to 300).
Equilibrium is (i guesses 300, j guesses 150)
eig | 300 - 1.5 150 | 75
Player i’s payoff is
S g (300,150) max{0,200 - eig }  max{0,100 - eig /10}
S g (300,150)  max{0,200  75}  max{0,100 - 7.5}  217.5
EXPERIMENTAL DESIGN
TWO-PERSON GUESSING GAMES
Within this structure, we vary the targets and limits independently and across
players and games, with targets either both less than one, both greater than one,
or mixed (the targets in previous guessing experiments varied only across
treatments, or not at all).
LOWER AND UPPER LIMITS:
(a) [100,500]; (b) [100,900]; (g) [300,500]; (d) [300,900].
TARGETS:
(1) 0.5; (2) 0.7; (3) 1.3; (4) 1.5.
Example (d4b1):
p i  1.5
300
100
p  0.5
j
900
900
EXPERIMENTAL DESIGN
TWO-PERSON GUESSING GAMES
Example (d4b1)
EXPERIMENTAL DESIGN
FEATURES OF TWO-PERSON GUESSING GAMES
• Reducing n to two allows us to focus on the central strategic problem of
predicting the guesses of other players who view themselves as a nonnegligible part of one’s own environment. It also eliminates his need to predict
how his guess affects an average.
• Having player-specific limits and targets, moves equilibrium guesses away
from the boundaries allowing clearer inferences.
• Our games are not zero-sum, and have more than two possible payoffs.
Consequently, players’ best responses to their beliefs are usually unique within
their limits. Deviations are costly.
EXPERIMENTAL DESIGN
TYPES
• L1 – plays the best response to beliefs that assign equal probabilities to
other's actions.
• L2 – plays the best response to L1.
• L3 – plays the best response to the best response to L1.
• D1 – which does one round of deleting actions that are dominated by pure
actions and then best responds to a uniform prior over other's remaining
actions.
• D2 – which does two rounds of deleting actions that are dominated by pure
actions and then best responds to a uniform prior over other's remaining
actions.
• EQUILIBRIUM – always plays his equilibrium action.
• SOPHISTICATED – plays the best response to the probabilities of his
partner's actions, which we estimate from the observed frequencies (depends
on data).
EXPERIMENTAL DESIGN
• (B) Baseline Treatment - Subjects played 16 guessing games, which are
generally asymmetric and dominance-solvable in 3 to 52 rounds, with
essentially unique equilibria determined (but not always) by players’ lower
limits when the product of their targets is less than one, or their upper limits
when the product is greater than one.
- We have eight player-symmetric pairs of games (so no need to subdivide
subjects into roles), and one pair of symmetric games.
- Repeated anonymous random pairing.
- No feedback about their own payoffs, their partners’ guesses and payoffs.
- All subjects framed as Row players and called “You”.
- Subjects were paid for 5 of the 16 games played. They were paid $0.04 per
point. With possible payoffs of 0 to 300 points per game, this yielded
payments from $0 to $60. Show-up fee of $8. Average total earnings were
$40.21.
EXPERIMENTAL DESIGN
Computer Interface
EXPERIMENTAL DESIGN
Click to open a box
EXPERIMENTAL DESIGN
Click to close the box previously opened
EXPERIMENTAL DESIGN
Entering a guess
EXPERIMENTAL DESIGN
• (OB) Open Boxes Treatment - is identical to the Baseline treatment except
that the 16 games are presented with the targets and limits continually visible,
in “open boxes”.
- Provides a test of whether interface affects guesses. We find insignificant
differences between Baseline and Open Boxes subjects’ guesses.
• (R/TS) Treatments - are identical to the Baseline treatment, except that each
subject is trained and rewarded as a specific type: L1, L2, L3, D1, D2, or
Equilibrium. Validates the model of players’ cognitive processes for the L1,
L2, L3, D1, D2, and Equilibrium types.
- Study the extent to which cognitive processes differ in the Baseline and
Robot/Trained Subjects treatments. The distribution of guesses of the Baseline
treatment is significantly statistically different from the distribution of guesses
of each one of the R/TS treatments.
EXPERIMENTAL DESIGN
DATA
- Guesses (Actions).
- Look-ups (box, gaze time - amount of time the box was "open").
Camerer, Johnson, Rymon, and Sen (1993) and Johnson, Camerer, Rymon,
and Sen (2002) pioneered the use of MouseLab data (actions, and information
search) in games by studying backward induction in extensive-form alternatingoffers bargaining games in which subjects could look up the sizes of the “pies”
to de divided in each period.
Costa-Gomes, Crawford, and Broseta (2001) used MouseLab to study twoperson matrix games, with unique equilibria.
Several other papers study other kinds of games and single-person decision
problems.
EXPERIMENTAL DESIGN
FEATURES OF THE DESIGN
• Varying targets and limits within an intuitive structure facilitates teaching
subjects the rules and allows them to concentrate on predicting others’
guesses and identifying best responses, which greatly reduces the noisiness
typical of initial responses to games.
• Types’ predicted guesses are more separated than in earlier studies, which
mostly used payoff matrix games, in which players have at most 5 possible
actions. In particular, the Lk, and Dk types are not separated in previous
guessing games experiments, and are only weakly separated in other
experiments.
• Allowing subjects to search within the intuitive common structure of
guessing games makes mental models of others easy to express as functions
of the targets and limits, as we will see next.
• As will also be seen next, the L1 type, which seems to best describe most
subjects’ actions in earlier studies, requires subjects looking-up elements of
the game about the other player (the lower and upper limits). In matrix
games, L1 subjects only need to look-up their own payoffs.
Computer Interface
(as in Costa-Gomes, Crawford, and Broseta (2001))
ECONOMETRIC MODEL
GUESSES-ONLY ANALYSIS
- Counting “Beans”
- Maximum-Likelihood Error-rate Analysis
- Specification Test (Overfitting, Omitted types)
SEARCH-ONLY ANALYSIS
- Maximum-Likelihood Error-rate Analysis
GUESSES AND SEARCH ANALYSIS
- Maximum-Likelihood Error-rate Analysis
- Specification Test (Overfitting)
ECONOMETRIC MODEL
GUESSES-ONLY ANALYSIS
Counting “Beans”
43 of 88 subjects made b/w 7 and 16 of some type’s exact (within 0.5)
guesses, 20, 12, and 3 conforming closer to L1, L2, and L3, than to Eq. (8).
Maximum-Likelihood Error-rate Analysis: We use a simple “spike-logit”
error structure in which, in each game (g), a subject (i) makes his type’s guess
exactly (t gk )with probability (1   ) , and otherwise ( ) makes errors, that follow
a logistic distribution over the rest of the interval between her/his limits.
U gik  [t gk  0.5, t gk  0.5]  [agi , bgi ]
Vgik  [agi , bgi ]/U gik
Expected payoff of guess xig given type-k’s beliefs:
Skg ( Rgi ( xgi ))
1000
  Sg ( Rgi ( xgi ), y ) f gk(y)dy
0
ECONOMETRIC MODEL
GUESSES-ONLY ANALYSIS
Maximum-Likelihood Error-rate Analysis:
The probability density function of an error (a guess that differs from type k’s
predicted guess) follows a logistic distribution over the rest of the interval
between the player’s lower and upper limits.
d gk ( Rgi ( xgi ),  )
exp(S gk ( Rgi ( xgi ))

k
Vgik exp[S g ( z )]dz
for Rgi ( xgi ) Vgik , and 0 elsewhere
We assume that, given type, errors in guesses are independent across games.
Subject i’s guesses related log-likelihood is:
L ( ,  | R ( x )) 
i
i
i
k
( G  nik ) nik
K
ln[ k 1 p (1   )

 gN ik d g ( Rg ( xg ),  )]
k
i
i
SOME FINDINGS
GUESSES-ONLY ANALYSIS
Maximum-Likelihood Error-rate Analysis (cont.):
Estimated Types: 43 L1, 20 L2, 3 L3, 5 D1, 14 Eq., and 3 Sop.
Hypotheses Tests:
-   1 is rejected for all but for 7 subjects. Spike is necessary.
-   0 is rejected for 34 subjects. Thus, the logit model’s payoff-sensitive
errors significantly improve the fit over a spike-uniform model for about 1/3 of
the subjects.
-   0,   1 which corresponds to a random model of guesses within our
specification is rejected at the 5% level for all but 10 subjects (6 L1, 2 D1, 1
Eq., and 1 Sop.)
ECONOMETRIC MODEL
GUESSES-ONLY ANALYSIS
“SPECIFICATION TEST”
Type estimates could be sensitive to our a priori specification of possible
types, which might err by omitting relevant types and/or overfitting by
including empirically irrelevant ones.
Test is based on the idea of a pseudotype - a pseudotype is constructed from
one subject’s guesses in the 16 games. Since we have 88 subjects, we are going
to have 88 pseudotypes. (Not all different, since if two subjects’ guesses
coincide in all games, their pseudotypes coincide).
We can then take some other subject’s guesses in the 16 games, and compute
the likelihood of the subject’s guesses, given a pseudotype’s (predicted)
guesses. (Not the same subject, since, obviously, the likelihood of a subject’s
guesses given the pseudotype based on its guesses is 1!)
ECONOMETRIC MODEL
GUESSES-ONLY ANALYSIS
“SPECIFICATION TEST”
The test compares the likelihood of our type estimate to the likelihoods of
analogous estimates based on 87 pseudotypes.
Omitting – Suppose we had omitted a relevant type, say L2. The pseudotypes
of subjects now estimated to be L2 would then outperform the non-L2 types
estimated for them, and would also make approximately the same (L2) guesses.
We define a cluster as a group of 2 or more subjects such that:
(i) each subject’s pseudotype has higher likelihood than the estimated type for
each other subject in the group;
(ii) subjects’ pseudotypes make “sufficiently similar guesses”.
Finding a cluster should lead us to diagnose an omitted type, and studying the
common elements of its’ subjects guesses may help to reveal its decision rule.
We find 5 clusters with 3, 2, 2, 2, and 3 subjects, respectively (see Table IX).
ECONOMETRIC MODEL
GUESSES-ONLY ANALYSIS
“SPECIFICATION TEST”
Overfitting – A subject’s estimated type to be a credible explanation of his
behavior should perform at least as well against the pseudotypes as it would,
on average, at random. Then for a pseudotype to have higher likelihood than
our estimated type it must come first among our 7 types plus itself, which has
probability 1/8. The subject’s estimated type has higher likelihood than all but
an expected number of 10.75 pseudotypes.
(15 type estimates are ruled out on the basis of overfitting – 10 L1, 2 L2, 1 D1,
1 Eq., and 1 Sop., of which 4 L1, 1 D1, 1 Eq., and 1 Sop. already ruled out on
the basis of a random model of guesses.)
SOME FINDINGS
SUMMARY OF REVISED GUESSES-ONLY ANALYSIS
Combining our guesses-only estimates with our statistical tests, we say that a
guesses-only type estimate appears reliable if:
(i) it does significantly better at the 5% level than a random model of guesses
within our specification.
(ii) it has higher likelihood than all but at most a random number of
pseudotypes.
(iii) it is not a member of any cluster.
By these criteria, 58 of our 88 subjects’ guesses-only type estimates appear
reliable.
SOME FINDINGS
SUMMARY OF REVISED GUESSES-ONLY ANALYSIS
L1: 43 (27 reliably identified, other 16 may be spurious - 5 in clusters + 11 no
better than random guesses and/or pseudotypes).
L2: 20 (17 reliably identified, other 3 may be spurious - 1 in clusters + 2 no
better than random pseudotypes).
L3: 3 (1 reliably identified, other 2 may be spurious - in clusters).
D1: 5 (1 reliably identified, other 4 may be spurious - 2 in clusters + 2 no
better than random guesses and/or pseudotypes).
Eq.: 14 (11 reliably identified, other 3 may be spurious - 2 in clusters + 1 no
better than random guesses and/or pseudotypes).
Sop.: 3 (1 reliably identified, other 2 may be spurious - no better than random
guesses and/or pseudotypes).
Our findings are close to previous estimates from other kinds of games.
ECONOMETRIC MODEL
Each type is associated with algorithms that describe how to process
information about targets and limits into guesses. We use these algorithms as
models of each subject’s cognition.
We infer a type’s minimal search implications from plausible algorithms for
identifying its ideal guess, under conservative assumptions about how cognition
affects search, like in our previous paper.
Standard assumptions imply a type looks up all freely available information that
may affect its beliefs and guesses its target times the mean of its beliefs.
The algorithms comprise basic operations and other operations.
We assume that basic operations are associated with adjacent look-ups, which
can appear in any order but cannot be separated; other operations can appear
in any order, and can be separated (although we report the most natural order,
we do not insist on it).
ECONOMETRIC MODEL
L1
i
j
j
p
(
a

b
)/2 ( 750) Search Implications {[ a j , b j ], p i }
Ideal Guess
ECONOMETRIC MODEL
L2 – 1st Step
j i
i
p
(
a

b
)/2 ( 300)
Her Guess
i i
j
([
a
,
b
],
p
)
Search Implications
ECONOMETRIC MODEL
L2 – 2nd Step
My Guess
p i R (a j , b j ; p j (a i  bi )/2) (  450)
Search Implications
{([ a i , bi ], p j ), a j , b j , p i }
ECONOMETRIC MODEL
SEARCH-ONLY ANALYSIS
Example of a Look-up Sequence: {ai, pi, pj, ai, bi, aj, pi, bj,…}
L2’s Look-up Sequence: {([ a i , bi ], p j ), a j , b j , p i }
Our econometric analysis quantifies compliance with a type’s search
implications as the density of the type’s relevant look-up sequence in the
subject’s look-up sequence.
Since subjects vary widely in where the relevant look-ups tend to be located in
their sequences, we filter out some idiosyncratic noise using a binary nuisance
parameter called style (“early” or “late”), (assumed constant across games) s  e, l
If s ,we
e start at the beginning of the subject’s look-up sequence, and continue
until we obtain the type’s complete relevant sequence:
Example: {ai, pi, pj, [ai, bi ], aj, pi, bj,…} (the L2’s relevant sequence has length
six, and the first complete sequence is obtained after eight look-ups,
compliance is 0.75).
ECONOMETRIC MODEL
SEARCH-ONLY ANALYSIS
We discretize compliance into three categories:
c  H , M , L ; CL  [0,0.333], CM  [0.333,0.667], CH  [0.667,1]
Let  c be the probability that subject i has type-k style-s compliance c in any
given game.
Let mcisk be the number of games for which subject i has type-k style-s
compliance c.
Subject i’s information search related log-likelihood is:
isk
Li ( p, s, | M isk )  ln[ kK1 p k s e,l I s  ( c )mc ]
c
58 out of 71 Baseline subjects’ style estimates are “early”, 10 are “late”, and 3
ties.
ECONOMETRIC MODEL
GUESSES AND SEARCH ANALYSIS
A subject’s type and style determine his information search and guess, each
with error.
We assume that, given type and style, errors in search and guesses are
independent of each other and across games.
Subject i’s information search and guesses log-likelihood is:
Li ( p, s, ,  ,  | M isk , N isk , Ri ( xi )) 
 K k

mcisk
( mcisk  ncisk )
ncisk
k
i
i
 ln k 1 p  I s [ ( c ) (1   )
( )
 d g ( Rg ( xg ),  )]
s e,l
c
gN cisk


The model has 6 parameters per subject: error rate  , precision  , type k ,
style s , and two independent probabilities  c .
ECONOMETRIC MODEL
GUESSES AND SEARCH ANALYSIS
Our subjects’ type estimates change when search is taken into account, for one
of two reasons:
- For some subjects there is a tension between guesses-only and search-only
type estimates, resolved in favor of a type other than the guesses-only estimate.
- For other subjects the type estimate based on guesses-only has 0 search
compliance in 8 or more games, and is therefore ruled out by a priori
constraints.
When the guesses-and-search type estimate differs from the guesses-only
estimate, we favor the former but require it to pass the analogs of the guessesonly criteria. We say that a guesses-and-search type estimate is reliable if:
(i) it does significantly better at the 5% level than a random model of guesses
and search within our specification.
(ii) The guesses-only part of its likelihood has higher likelihood than all but at
most a random number of pseudotypes.
(iii) it is not a member of any cluster.
SOME FINDINGS
SUMMARY OF GUESSES AND SEARCH ANALYSIS
The search and guesses analysis:
- Confirms the reliability of 39 of our 46 Baseline subjects whose guesses-only
type estimates appear reliable.
- It also reliably identifies 2 subjects as L1 who had appeared reliably
identified as L2.
- Confirms the reliability of 1 L1 and 1 L2 subjects whose guesses-only
estimates were inconclusive.
- It calls into question the type estimates of 4 subjects who had appeared
reliably identified: 1 L1, 1 L2, 1 D1, 1 Eq.
In short, 54 of 88 subjects’ types are identified with confidence: 29 L1, 14 L2,
10 Eq., 1 L3, 1 Sop. (last was OB, and might not have survived monitoring
search).
SOME FINDINGS
• IDENTIFIED TYPES AND DECISION TIMES: average time per guess
according to subjects’ estimated types (based guesses and information
search).
B Treatment:
L1 (22): 63.7 sec.
L2 (13): 82.1 sec.
Eq. (8): 117.2 sec.
CONCLUSIONS
• Reports an experiment that elicits subjects’ initial responses to 16 dominancesolvable two-person guessing games, monitoring their searches for hidden but
freely accessible parameters of the games along with their guesses.
• The design yields strong separation of the guesses and information searches
implied by leading decision rules.
• Cognition is studied at the individual level as in Costa-Gomes, Crawford,
and Broseta (1998) UCSD-DP.
• Of our 88 Baseline and OB subjects, 54 can be reliably identified as one of
our types based on guesses and search.
• Because our types specify precise guesses in large strategy spaces, the
identifications show that those subjects have accurate models of the games and
acted as rational, self-interested expected-payoff maximizers.
• Lk types are overwhelmingly predominant. This lends support to the leading
role given to iterated best responses in informal analyses of strategic behavior.
EXPERIMENTAL DESIGN
TREATMENTS
• BASELINE (B) vs. OPEN BOXES (OB): provides a test of whether
interface affects guesses and decision time.
Time to make a guess increases (from 74.50 to 91.87 seconds).
The distribution of guesses are not significantly statistically different.
• BASELINE (B) vs. ROBOT/TRAINED SUBJECTS (R/TS): validate
the model of player’s cognitive processes for the L1, L2, L3, D1, D2 and
Equilibrium types; study the extent to which cognitive processes differ in the
Baseline and Robot/Trained Subjects treatments.
Time to make a guess decreases (from 74.50 to ??? seconds).
The distribution of guesses of the Baseline treatment is significantly
statistically different from the distribution of guesses of each one of the
R/TS treatments.