Why Are Two Heads Better
Than One?
David J. Cooper
Florida State University
John H. Kagel
Ohio State University
Eric Mayefsky
Stanford University
Background/Prior Research


Cooper and Kagel (2005) report that two-pwerson teams outperform
individuals in terms of developing substantially higher levels of
strategic play than individuals in a signaling game (a stylized version
of Milgrom and Roberts, 1982, entry limit pricing game).
Not only the teams have higher levels of strategic play, but they beat
a demanding Truth Wins (TW) norm that a team containing N
individuals does better than N individuals operating independently.



Performance of teams relative to TW is particularly strong when Ss are asked to
take what have learned in one game and apply it in a second, related game.
Psychology research on team play in eureka type intellective
problems provides ample evidence that teams typically fail to beat
the TW norm (Kerr and Tindale, 2004). Nevertheless ours do.
KT note that in cases where the team play beats the TW norm, such
claims should be examined carefully. The present research is
aimed at better understanding why teams do so well in CK (2005).
Why Might Teams Do Better than Individuals in Limit-Price
Game

Being member of a team affects individuals’ ability to reason about
the game without the need for bilateral interactions.

Individuals’ abilities to reason may be stimulated by having to explain
how the game is played, consistent with the extensive psychology
literature on accountability (Lerner and Tetlock, 1999).



Being accountable to others can, in certain circumstances, promote selfcritical and effortful thinking.
Having to interpret a teammate’s discussion may stimulate individual’s
own cognitive processes.
Bilateral communication inherent in operating as a true team may
help generate insights that would not have occurred otherwise.

Teamwork may play a direct role in generating the strong performance
of teams so that bilateral interactions are necessary for teams to beat
the truth wins criterion.
Two new experiments designed to differentiate these hypotheses.

Experiment 1 replaces freely interacting two-person teams with
pairings of an adviser and advisee who choose actions separately
Bilateral communication is replaced by one way communication
from an adviser to their advisee.


Advisees have the benefit of “two heads,” their own and their advisors,
but no scope for interactive discussions. If bilateral communication is
critical to strong performance of teams, advisees will not perform as well
as teams in learning to play strategically. This holds even more so for
advisers who do not have “another head” to work with.
Experiment 2 looks at the effect of previous experience in fully
interactive teams on cross-game learning when playing as
individuals.

Our teams research suggests that team play promotes the development
of strategic reasoning. Experiment 2 addresses the questions of
whether team play is necessary to trigger this strategic empathy in a
related game.
Design of Experiment 1 – Two Treatments


Games with low cost Es throughout – only pure strategy equilibrium
is separating. Play converges to the efficient separating equilibrium
with low cost Ms limit pricing.
Cross-Over Games

Inexperienced Ss play games with only high cost Es present which
permits pure strategy pooling equilibria with high cost Ms limit pricing
(which play reliably converges to).
 Experienced sessions replace high Es with low cost Es, which destroys
the pooling equilibrium.
 In a deep sense strategic play following the cross-over is the same as
before – making Es believe you are the type of M who will be a tough
competitor. However, the actions needed to carry this out are quite
different. With pooling high cost Ms imitating low cost types. With
separating low cost Ms need to distinguish themselves from high cost
Ms.
The Limit Pricing Game (Milgrom and Roberts (1982)
M’s Payoffs as a Function of E’s Choice
MH
E’s Choice
In
Out
150
426
168
444
150
426
132
408
56
182
-188
-38
-292
-126
Output
1
2
3
4
5
6
7
Output
1
2
3
4
5
6
7
ML
E’s Choice
In
Out
250
542
276
568
330
606
352
628
334
610
316
592
213
486
B’s Payoffs
High Cost Entrants
B’s Choice
In
Out
M’s Type
MH
300
250
ML
74
250
Expected
Payoffa
187
250
Low Cost Entrants
B’s Choice
In
Out
a
M’s Type
MH
500
250
This information was not provided as part of payoff tables.
ML
200
250
Expected
Payoffa
350
250
Advice Treatments and Procedures





Chat is one way
Pairings fixed within a session but not between inexperienced and
experienced sessions.
Yoked advisers-advisees – so same M types (but face different Es).
Public information issues – two sets of sessions.
Advisers get extra payment equal to 30% of advisees earnings.


Inexperienced subject sessions




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24 games, two 12 game cycles.
Experienced subject sessions


Not taken out of advisees earnings.
32 games, four 8 game cycles.
Random Matching (Strangers)
Role Switching
Generic Context
Ss paid for all rounds.


1 franc = $.0025 + $20 show up fee. (Use of “double headers”)
Average sessions earnings $23 ($30) inexper (exper) sessions.
Low Cost Entrant Games, Experienced
Subjects
Strategic Play by MLs, Low Cost Entrant
Games
1.0
% Strategic Play
0.8
0.6
0.4
0.2
0.0
Inexperienced
Cycle 1
Inexperienced
Cycle 2
1x1
2x2
Experienced
Cycle 1
Advisor
Experienced
Cycle 2
Advisee
Experienced
Cycle 3
Experienced
Cycle 4
Simulated 2 x 2s, Median
Strategic Play by MLs, Crossover Sessions
1.0
% Strategic Play
0.8
0.6
0.4
0.2
0.0
Crossover
Cycle 1
1x1
Crossover
Cycle 2
2x2
Advisor
Advisee
Crossover
Cycle 3
Simulated 2 x 2s, Median
Possible Role of Accountability Effects in
Strong Play of Advisers
 Accountability:
“Self-critical and effortful thinking is
most likely to be activated when decision makers learn
prior to forming any opinions that they will be accountable
to an audience (a) whose views are unknown, (b) who is
interested in accuracy, (c) who is interested in processes
rather than specific outcomes, (d) who is reasonably well
informed, and (e) who has a legitimate reason for
inquiring into the reasons behind participants'
judgments.” (Lerner and Tetlock, 1999)
Content analysis of Advice


Chat levels are much lower (and less informative) with advice than
with teams (3.4 vs 18 relevant messages in 24 rounds as inexperienced Ss).
Do explanations matter?
Given Explanation for Strategic Play
Advised to Play
Strategically as
an ML
No
Yes
No
.298
(168)
.375
(8)
Yes
.525
(40)
.620
(73)
Suggests that primary effect of advice is to make Ss consider strategic
play. Game theoretic models virtually always assume that all available
strategies are considered at all times. In reality Ss may simplify the
game by not considering some strategies, locking on to one or two
alternatives. Advice to play strategically can then get advisees to
consider strategies that they have ignored.
Further Analysis – Maybe One Can Do Better Than With
Teams

Absent process loss in teams
π (2x2) = 1 – (1-p(adviser))2 > p(adviser)
where π (2x2) is observed prob of strategic play by MLs in teams
and p (adviser) is prob of strategic play by advisers as MLs.


That is, advisers not predicted to play as strategically as teams as they
do not have a “second head” to work with. We check for this.
Absent process loss in teams
π(advisee) = 1- (1-p(adviser))*(1-p(1x1)) > p(1x1)
where p(1x1) is the prob of strategic play in the 1x1 sessions.
That is, advisees are predicted to play at the level of a simulated team
made up of one advisee and one 1x1 player where TW. We check for
this.


Simulated teams of advisers meet or beat real teams games with
low cost Es throughout and beat real teams in all crossover cycles.
Simulated teams of advisers and 1x1 players tend to meet or beat
levels of strategic play of advisees which, absent team process loss,
should not happen. Why?
Advisers do not always provide advice even when they play strategically –
provide it only 68% (76%) of time as inexperienced (experienced) Ss. Reducing
p(adviser) by these amounts alone accounts for the drop-off. This cannot be free
riding or due to excess chatter, just a reluctance (or lack of confidence) to
express thoughts.
 Advisees only play strategically, after being given advice to do so, 63% (91%) of
time for inexperienced (experienced) Ss. So truth does not always win!
 But advisees are clearly doing quite well, on their own, as in the first cycle
following the crossover, around half of the advisees got not advice, but still
played strategically 60% of the time, which is a bit higher than teams in the same
time frame.


This suggests that there might be a better way to organize than in
teams. Similar to Armstrong’s (2006) results for face-to-face
communication in judgmental tasks.
Experiment 2



Uses cross-over treatment only.
Ss first play in teams in games with high cost Es that converge to
strong pooling equilibrium.
They then return to play as individuals – one cycle as high cost Es,
after which introduce low cost Es where the only pure strategy
equilibrium involves separating.
Figure 8: Strategic Play by MLs, Crossover Sessions
1.0
% Strategic Play
0.8
0.6
0.4
0.2
0.0
Crossover
Cycle 1
1x1
2x2
Crossover
Cycle 2
Advisor/Advisee
2x2→1x1
Crossover
Cycle 3
Modified Truth Wins Benchmark
Conclusions

Bilateral communication is not necessary to generate
high levels of strategic play in teams. Individuals are
better in teams rather than teams being better than
individuals.
Strategic play in advisor/advisee sessions is virtually the same as
in team sessions. Possible role for accountability effects.
 Alternative explanation that needs to be checked out – “selfadvice.”

Results suggest there might be better organizational
structures than teams – as unaccounted for process
losses.
 The strategic sophistication that team play develops
carries over to individual subject play.
 To the extent team play is designed to identify, through
analyzing chat records, the underlying behavioral
processes at work in strategic interactions, teams are a
far better medium for doing so compared to advice.
