We compare behavior in a one-shot Centipede game across several payoff structures including nonlinear payoff tournaments. Assuming Nash to be optimal, results suggest nonlinear tournament payoffs based on overall relative rewards are not sufficient to increase Nash results in the one-shot Centipede style setting. Evidence suggests that reducing strategic uncertainty is more important than increasing dominance in promoting Nash play.Centipede game, payoff tournaments, experiment, strategic uncertainty, dominance

Andrew J Moravec

Christopher R Mcintosh

Jason F Shogren

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 3 
  
Can tournaments induce rational play in the centipede game? Exploring 
dominance vs. strategic uncertainty  
  
 
 
Christopher R Mcintosh  
University of Minnesota Duluth 
Jason F Shogren  
University of Wyoming 
Andrew J Moravec  
University of Minnesota Institute of Technology
Abstract 
We compare behavior in a one-shot Centipede game across several payoff structures including nonlinear payoff 
tournaments. Assuming Nash to be optimal, results suggest nonlinear tournament payoffs based on overall relative 
rewards are not sufficient to increase Nash results in the one-shot Centipede style setting. Evidence suggests that 
reducing strategic uncertainty is more important than increasing dominance in promoting Nash play.
Thanks to the Stroock and Bugas endowments for financial support and Professors Kenneth Gerow, Robert Godby, and an anonymous 
referee for advice. 
Citation: Christopher R Mcintosh and Jason F Shogren and Andrew J Moravec, (2009) ''Can tournaments induce rational play in the 
centipede game? Exploring dominance vs. strategic uncertainty '', Economics Bulletin, Vol. 29 no.3 pp. 2018-2024. 
Submitted: May 27 2009.   Published: August 18, 2009. 
 
     1.  Introduction 
 
Recent work has shown that institutions which spotlight the opportunity costs of 
irrationality can overcome seemingly anomalous behavior (e.g., Cherry et al. 2003, Cox and 
Grether 1997). In the wilds, one classic institution is the tournament (e.g., Ehrenberg and 
Bognanno 1990, Jensen and Murphy 1990). Tournaments increase incentives for rational play 
because they provide sizable rewards for small/flat differences in measurable performance, 
thereby addressing one of the main problems plaguing gaming experiments for decades (e.g., 
Harrison 1989).  In our previous experimental work on the power of institutions to generate 
rational behavior, we found that adding tournament incentives into classic gaming experiments 
resulted in more rational gaming behavior, not less, as has been feared by some (see Camerer and 
Hogarth 1999). We observed more self-interested bargains, more subgame perfection outcomes 
in endogenous timing contests, and more sincere bidding behavior in second-price auctions (e.g., 
Shogren et al. 2006, Baik et al. 1999, Shogren 1997).    
 
Herein we want to up the ante—to stress test the power of tournament incentives to 
induce similar patterns of rational play in an unfavorable gaming environment, the one-shot 
centipede game. The one-shot centipede game provides a good robustness test of the power of 
tournament incentives because people rarely play as predicted—few players end the game 
immediately as predicted by backward induction that eliminate dominated outcomes, i.e., an 
immediate “take” (see Figure 1; Rosenthal 1981); Rather experimental evidence suggest a 
dominance problem exists. Unmotivated players seem unable or unwilling to work through more 
than a few steps of iterated elimination of dominated strategies (McKelvey and Palfrey 1992). 
While some dominance-solvable gaming experiments confirm that stronger incentives can help 
him over come his own dominance problem, he can still be affected by strategic uncertainty. 
Strategic uncertainty exists if he cannot safely presume the other players will also behave 
rationally or that they will believe he will act rational (see review in Camerer 2003). We consider 
two non-linear payoff tournaments—relative reward rankings and within game relative 
rankings—and these structures relationship between dominance and strategic uncertainty in 
inducing more people to act as if they were rational.   
 
2.  Experiment 1: Relative reward tournament 
 
Participants were recruited from undergraduate economics classes for two sessions, each 
with N = 21 subjects. Monitors read the basic information and the instructions for the five payoff 
structures; called cases  (experimental instructions available on request). The five case 
instructions were randomly arranged within a packet of instructions for each player. They 
received a $5 participation fee upfront, and were notified that one case would be chosen at 
random to determine their take-home earnings. Each case described how points would be 
converted into dollar earnings.  
 
 
 
 
 
 
  1  
Figure 1: Point Tree 
Player A  Player B  Player A  Player B  Player A  Player B 
Stage 1   Stage 2   Stage 3   Stage 4   Stage 5   Stage 6 
h   h   h   h   h   h      (256, 64) 
       Pass       Pass      Pass       Pass     Pass         Pass   (1, 0) 
                
     Take        Take        Take        Take       Take                   Take 
 
E1 (4, 1)*   (2, 8)     (16, 4)   (8, 32)             (64, 16)            (32, 128) 
E2 (1,0)**   (0, 1)     (1,0)    (0,1)             (1,0)                 (0,1) 
*(Player A’s points, Player B’s points) 
**(Player A’s tallies, Player B’s tallies) 
 
Figure 1 shows the centipede game used in the experiment, called the point tree, which 
was identical for all games (in Figure 1 see first row of payoffs, E1). Each subject knew he or 
she would play one game against each and every other subject in their session (N – 1 = 20 
games) per case to determine payoffs. Subjects were informed their role (Player A or B) in each 
game would be determined randomly. As required by games of complete and perfect 
information, every subject specified his complete strategy space by filling out one recording 
sheet for the five cases. The sheet asked whether he would PASS or TAKE at each of the six 
decision stages. Once the recording sheets for each of the five cases were completed, the subjects 
were finished with their task. Payoffs were determined and the subjects were paid in cash.  
 
Experiment 1 considered five payoff structures, which were presented to the subjects as 
colors to remove economic jargon. The tournament with nonlinear payoffs (Blue) used a relative 
ranking scheme:   
  $180   Winner—player  with  greatest total points at the end 
$100   2
nd  
$60   3
rd  
$25   4
th  
$12   5
th-10
th 
$7   11
th – 21
st  
 
We designed the tournament with a $180 winner-payoff for less than an hours work  to get all 
players to pay more attention to the incentives at work in the game, i.e., removes the temptation 
to PASS given the substantial opportunity cost for not eliminating dominated outcomes. If this 
reward structure is sufficient, each player will end the game immediately because he believes all 
other players will also follow the same rational strategy, which eliminates strategic uncertainty. 
But if this reward is insufficient, some players might be tempted to PASS, say in stage 1, hoping 
his opponent does the same in 2, so he can TAKE in stage 3, outscoring his opponent by more 
points (e.g., 12 = 16 – 4 vs. 3 = 4 – 1), which increases his total points and ranking.   
  
As comparative benchmarks to more classic incentives, we consider four additional 
payoff structures: three base-bonus payoffs (Green, Orange, Pink) and one nonlinear step 
function (Purple). See Table I (below) for descriptions. 
  2Now consider the results. A computer program combined each player’s strategy against 
the other 20 players, in every combination of subjects and Player A/B roles (21 players playing 
each other twice = 420 games). We define our results by the “terminal” stage —the stage in 
which a player chooses TAKE for the first time.
 Table I (column E1) summarizes the stage 1 
results.  
 
Table I: Payoff Structures and Results 
Payoff Structure 
Expected 
Nash Pay 
Max 
Pay 
E1 Stage 1 
Ending  
Proportion
E2 Stage 1  
Ending 
Proportion 
Difference 
Test  
(p- values) 
Nonlinear Tournament (Blue) 
See payoffs in paper 
$24.48   $180  0.119  0.381  p = 0.027 
         
High Base / Low Piece Rate 
(Green) 
$16 plus 1.5¢/cumulative point 
$16.75   $93  0.167  0.167  p = 1.00 
         
Medium Base & Piece (Pink) 
$8 plus 3¢/c pt 
$13.00   $62  0.143  0.119  p = 1.22 
         
Low Base / High Piece (Orange) 
5¢ /c pt 
$9.50   $161  0.071  0.119  p = 0.524 
         
Nonlinear Step (Purple) 
$10-62; based on point ranges 
$2.50  $62  0.095  0.023  p = 1.82 
Points;               Payoff         
20-40;               $10         
41-60;               $13         
61-100;             $17         
101-200;           $22         
201-500;           $28         
501-1,000;        $35         
1,001-2,000;     $43         
2,001-3,500;     $52         
3,501+;             $62         
 
Despite the relatively large payoff that goes to the winner and runner-up ($180 and 
$100), the tournament did not outperform the other payoff structures at stage 1: only 11.9 percent 
rational play was observed, which fell between the 7.1 percent low in Low Base/High Piece and 
the 16.7 percent high for High Base/Low Piece. There was no statistical difference between any 
of the payoff structures for games ending in stage 1.   
 
  33.  Experiment 2:  Within-game relative rankings 
 
Why did players have the same response to the five payoff structures? (1) Was the 
environment too complicated for players or (2) were the incentives too weak to induce 
significant changes in rational behavior? We address these questions using a second tournament 
structure—within-game competition. In the wilds, within-game rewards are used to create 
stronger incentives then overall rewards, e.g., round robin sporting events in which wins and 
losses determine relative rankings and earnings (see Ehrenberg and Bognanno 1990). We use 
tallies to create within-game relative payoff rewards.  A tally is earned for the player with the 
most points in each game. If player A has more points than B in game 17, A receives 1 tally. 
Two sessions of 21 students participated.    
 
All aspects of Experiment 2 were identical to Experiment 1, except that we added the 
tallies into the Nonlinear Tournament. The winner who earned the most “tallies” would receive 
$180. The other four cases are identical to Experiment 1. The use of tallies transforms Figure 1 
payoffs E1 into payoffs E2 —a 0-1 binary game. Note players never saw this transformed tree; 
they were given Figure 1 with payoffs E1. 
 
Adding tallies transformed the centipede game into simple first-“taker” wins game. We 
recognize this is extreme—but we purposefully designed this experiment to test whether we can 
provide any incentive, extreme or not, for a majority of people to TAKE in stage 1 by only 
altering payoff structure to induce the Nash outcome. If most players respond to the tally 
tournament incentives, then we have some evidence that rational play in the one-shot centipede 
game is not universally rejected. 
 
  Our new results are positive—tallies tripled the level of subgame perfect play to 38.1 
percent (Table I E2). Tally behavior differed from the other four incentive designs at the 3% 
level. The last column of Table I compares behavior across experiments 1 and 2. The test 
between the original and tally tournament gave a two-tailed p-value of 0.0272. The other four 
cases did not differ across experiments. 
 
  Experiment 2 results allow questions (1) and (2) above to be addressed. There is evidence 
that the environment was not too complicated. Comparing Table I E1 and E2 illustrates that the 
only significant change in play occurred for the tournament payoff structure. There was strong 
consistency in the play of the other four structures (having not changed) across the experiments 
by the different subject pools. This suggests that subjects were responding to the nature of the 
payoffs and not randomly assigning strategies across structures. The results suggest the 
Experiment I tournament did not provide strong enough incentives to induce rational behavior. 
Strategic uncertainty was clearly not eliminated leading to a vast majority of subjects passing at 
stage 1 likely in hopes of earning increased points (recall this is rational if it is believed the 
opponent will PASS).  
 
When this temptation was removed (cannot earn more than one tally per game) several 
new respondents recognized that strategic uncertainty was not present and played accordingly, 
leading to a statistically significant change in the proportion of games ending in stage 1. These 
subjects were not victims of a dominance problem; their behavior is consistent with them having 
  4thought through how points were converted to tallies (recall the point tree they were given was 
not changed) and recognized the Nash equilibrium was also free of strategic uncertainty. This is 
further evidence that the environment was not too complicated, behavior adjusted to the change 
in incentives.  
 
4.  Discussion 
 
Can tournament incentives induce people to play rationally in the centipede game? Our 
results are mixed —less than 1 in 10 people played as predicted in a tournament setting; whereas 
nearly 4 in 10 did in the tally tournament. From the perspective of trying eliminating strategic 
uncertainty, we moved subjects toward rational play with the tally tournament. But the increase 
in rational play came at a significant cost in terms of institutional credibility—we contorted the 
game into an extreme version of the centipede game. If such strict institutions are need to induce 
more rational play, additional research is needed to better understand how to institute the right 
incentive instead of simply adding additional money.   
 
We purposefully chose a one-shot game to see if we could over come dominance just 
with serious incentives, not from learning by ones mistakes through repeated small payoff 
games.  One might step back from the extreme and still increase rational play with repetition and 
feedback, as some learning will occur. Shogren and Hurley (1997), for example, found some 
evidence of learning in a four-round elimination centipede tournament.  They observed rational 
play for 6 in every 10 players in the practice rounds, 7 in 10 in Round 1, and 10 in 10 in rounds 
2-4 of the tournament.  How much repetition is necessary for this to take place is a question for 
future research.   
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Can tournaments induce rational play in the centipede game? Exploring dominance vs. strategic uncertainty

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Can tournaments induce rational play in the centipede game? Exploring dominance vs. strategic uncertainty

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