Activity Bias and Focal Points in the Centipede Game

 
 
Activity Bias and Focal Points in the Centipede Game


                                            August 2011


                     Evren Atiker, William S. Neilson, Michael K. Price *




Abstract:
        Previous experiments in the centipede game have found extremely low
frequencies of the predicted subgame perfect equilibrium (SPE) play. This paper
explores why. By making small changes to the payoffs, but not the structure, of the basic
centipede game we are able to determine whether the observed propensity to continue the
game beyond the SPE node is driven by (i) a desire to increase joint payoffs, (ii) beliefs
about the rationality of one’s opponent, (iii) activity bias, and (iv) the lack of focal points.
Previous research on strategic form games has explored the influence of all of these
factors separately. Our experimental results rule out the influence of efficiency
considerations and beliefs about opponent rationality. Activity bias is a contributing
factor, but focal points provide the largest effect. The research therefore points to the
importance of a new consideration for dynamic games, and parses between factors that
move subjects away from subgame perfect play.

        Keywords: Centipede game; subgame perfect equilibrium; focal points; activity
              bias; backward induction
        JEL codes: C7, C9




*
 Atiker: Department of Economics, University of Tennessee, Knoxville. eatiker@utk.edu. Neilson:
Department of Economics, University of Tennessee, Knoxville. wneilson@utk.edu. Price: Department of
Economics, University of Tennessee, Knoxville and NBER. mprice21@utk.edu.

         We thank Kelly Padden Hall and P.J. Healy, seminar participants at Middle Tennessee State
University and the University of Tennessee, and attendees at the 2009 Tucson Economic Science
Association conference for helpful comments. Funding for the project was provided by the National
Defense Business Institute.

                                                                                                      0
1. Introduction

         Since its introduction by Rosenthal (1981), researchers have used the centipede

game to test for equilibrium behavior in a sequential move, complete information setting.

The centipede game itself consists of alternating play of binary choices. At each decision

node, the player making the choice must decide whether to end the game or continue by

passing to the next player. Continuing the game increases the total payoff to the two

players but switches who receives the larger payoff. Importantly, if player A chooses to

continue but B chooses to end the game at the very next node, A’s payoff is lower than it

would have been if she had ended the game at the previous node. This payoff structure

yields a single subgame perfect equilibrium (SPE) strategy combination – both players

elect to end the game at every choice node. Hence, equilibrium play prescribes the game

ending at the very first node.

         Yet, study after study finds that subgame perfection fails to organize behavior.

For example, McKelvey and Palfrey (1992) provide the first experimental test of the

centipede game and find SPE outcomes in 7.1 percent of their four-move games and only

0.7 percent of their six-move games.                 Subsequent studies have changed the basic

structure of the game and found similar frequencies of SPE play. Nagel and Tang (1998)

test a normal-form version of the game with 0.5 percent SPE outcomes. Parco, Rapoport,

and Stein (2002) and Rapoport et al. (2003) examine a three-player version of the game

and find SPE play in 2.5 percent and 2.6 percent of all respective games. 1 Bornstein,

Kugler, and Ziegelmeyer (2004) compare the behavior of individual decision-makers

with groups of three but find no SPE play in either treatment. Finally, Palacios-Huerta


1
  It should be noted that Rapoport et al. (2003) find significantly higher frequencies of SPE play as the
stakes of the game are increased.

                                                                                                            1
and Volij (2009) and Levitt, List, and Sadoff (2010) use a more sophisticated subject

pool, accomplished chess players, to explore the importance of rationality and the beliefs

about the rationality of others on SPE play. While the former find a very high (72.5

percent) frequency of SPE outcomes, the latter only find SPE play in 3.9 percent of all

games and no instances of SPE outcomes when the players are grandmasters.

        The purpose of this paper is to catalog different influences that could lead players

to deviate from equilibrium play and construct laboratory experiments to isolate and

measure the relative importance of these influences on observed play. 2 We consider four

distinct confounds that may explain the failure of backward induction in the standard

centipede game: (i) a desire to increase joint payoffs, (ii) beliefs about the rationality of

one’s opponent, (iii) activity bias, and (iv) the lack of focal points.

        Empirically, we provide the first apples-to-apples comparison of the relative

importance of these different factors. To facilitate such a comparison, we augment the

standard centipede game by removing (or enhancing) a particular confound and

observing how this affects departures from self-interested play. All of the changes we

consider are “small” in that they (i) make either minor or no changes to the strategy space

and (ii) with one notable exception, make no changes to the equilibrium path. Yet, they

enable us to parse different influences that drive play in the centipede game.

        The first possible confound we consider arises because the standard SPE does not

maximize the players’ joint payoffs. Researchers such as Charness and Rabin (2002) and

Engelmann and Strobel (2004) have devised models to account for such preferences and

the data are certainly supportive. We consider two adaptations of the standard centipede

2
 In this regard our approach builds upon Levitt, List, and Sadoff (2010) who combine data from
complementary experiments to explore whether deviations from equilibrium play are related to players’
ability to backward induction and/or their beliefs about the ability of others to do so.

                                                                                                        2
game to identify preferences for efficiency. The first approach follows Fay, McKelvey

and Palfrey (1996) by holding constant the joint payoff at all decision nodes.

Strategically, this constant sum centipede game is equivalent to the standard game

without growth in payoffs across nodes. Our second approach adds pairs of nodes to the

beginning of the centipede game for which continuing the game increases both individual

and joint payoffs. Yet, in all such “early move” games, the SPE outcome yields payoffs

identical to those from the standard centipede game.

         The second possible confound we consider arises as play may deviate from the

subgame perfect prediction if rationality is not common knowledge. Aumann (1995,

1998) details the extent to which common knowledge or rationality is required for

backward induction to occur. 3 For example, if player A believes that player B will play

choose to continue the game at her first node, then it is a best response to this belief to

continue the game at the initial node. Similarly, if player B believes that A will continue

the game at her second node, B’s best response is to continue the game at her first node

and allow this to happen. 4

         To isolate whether deviations from SPE play reflect beliefs about rationality, we

remove the final two nodes of the standard centipede game and insert a new pair of nodes

that provide one player an extremely large payoff and the other a correspondingly small

(sometimes negative) payoff. Importantly, players can only receive this large payoff

should their opponent end the game and receive a very small (negative) payoff. As

3
  Recent popular manifestations of Aumann’s ideas take the form of cognitive hierarchy theory (Camerer et
al. 2004) and level-k thinking (Stahl and Wilson, 1995; Nagel, 1995; Costa-Gomes and Crawford, 2006).
McKelvey and Palfrey (1992, 1995, 1998) exploit this notion in their analysis of centipede game data using
quantal response equilibrium.
4
  Of course, this raises the possibility that player A continues the game at the first node to manipulate B’s
beliefs about A’s rationality, and the complexity of the analysis begins. The seminal work on such issues is
Milgrom and Roberts (1982). Crawford (2003) and Hendricks and McAfee (2006) show how strategic
belief manipulation helps understand the D-Day invasion.

                                                                                                           3
subgame perfection and virtually every other behavioral hypothesis predict that players

will choose to continue the game at these nodes, any belief system that makes it a best

response for a player to continue the game at the initial node is likewise supported

following this change. 5 Thus, if beliefs about rationality are an important driver of

behavior, subjects should select continuing the game at their initial decision node with

greater frequency in these “rationality” games.

         The third potential confound we consider occurs as the subgame perfect strategy

precludes player B from having an influence over the game’s outcome and associated

payoffs – an outcome to which player A may be averse. This is an action bias and has

been documented in a wide variety of contexts (see, e.g., Patt and Zeckhauser, 2000; Lei

et al., 2001; Bar-Eli et al., 2007). To examine the role of activity bias, we augment the

standard centipede game by providing player B a trivial choice should player A select to

end the game at the initial node.

         Finally, play in the centipede game may be difficult because the game lacks focal

points as defined by Schelling (1960). In the standard centipede game, all nodes involve

the same tradeoff and are thus equally focal. Yet, if one of player B’s nodes was made

more focal, it could enhance backward induction – i.e., player A could better determine

player B’s action at that node and react accordingly. 6 To examine whether the failure to

fully backward induct in the centipede game is driven by a lack of focal points, we




5
  In fact, in all such games, the attractiveness to B of continuing the game at her first node is enhanced - she
is no worse off if A continues the game at the second node but is much better off if A selects to end the
game at this node.
6
  Evidence from coordination games suggests that the degree of “focalness” can be manipulated (Crawford
et al. 2008), and that focal points can work even if the point of focus is not an equilibrium (Bosch-
Domenech and Vriend, 2008).

                                                                                                              4
change the payoff disparity for either the second or fourth pair of decision nodes to make

these nodes more focal.

       Our empirical results call into question the first two explanations as important

determinants of behavior. For example, while the fraction of players ending the game at

the first node of the constant sum game increases fourfold, we observe a non-trivial

proportion of subjects (approximately 9.65%) ending the game before the SPE node in

our “early-move” games – i.e., ending the game while both players’ payoffs are still

growing. While the evidence from the constant sum game is suggestive of a preference

for efficiency, the evidence from the early-move games is inconsistent with the

hypothesis of joint-profit maximization.

       Evidence from our “rationality” games rejects the hypothesis that beliefs about

rationality influence play. Rather than observing a reduction in SPE play, the frequency

with which players end the game at their first node increases dramatically. For the A

player, the likelihood of stopping the game at the initial node increases three- to

sevenfold.

        Instead, our data suggest that play in the standard centipede game reflects a

combination of activity bias and the lack of focal points. For example, providing player

B a trivial choice if player A selects to end the game at the initial node triples the

frequency of SPE play for player A. Yet, the inclusion of this added choice has no

influence on player B’s choice at the initial decision node. Similarly, inserting a focal

point by changing the payoff disparity at a pair of nodes has a dramatic influence on

observed behavior. The introduction of an early focal point increases the frequency of




                                                                                        5
subgame perfect behavior by a factor of eight.                    We observe similar, albeit less

pronounced effects in games with late focal points.

2. Games and experimental design

       Figure 1 shows what we refer to in this study as the “standard” centipede game.

Two players, A and B, alternate play and at each node can choose either Down or Right.

Playing Down ends the game. Playing Right continues the game and has a uniform

impact on payoffs – it adds 2 to the larger payoff, adds 1 to the smaller payoff, and

switches which player gets the larger payoff. Centipede games thus have a particular

payoff structure that guarantees a unique subgame perfect equilibrium strategy

combination. Letting πi,t be the payoff to player i from a Down move at a node nt at

which i makes the decision and πj,t by the payoff to player j at that same node,

centipede payoffs satisfy



                                    πi,t + 1 < πi,t < πi,t + 2                                (1)

and

                                    πj,t < πj,t + 2 < πj,t + 1.                               (2)



Following Neilson and Price (2011), we define any sequence of nodes t to τ that adhere to

this payoff structure as the centipede chain. Using this definition, note that for our early

move games the centipede chain excludes the initial nodes of the game. Hence the

centipede chain for these games is shorter than the total length of the game as measured

by the total number of nodes.




                                                                                               6
Figure 1
                                  Standard Centipede Game

    A        B       A     B       A      B       A       B     A       B
                                                                               40,25


  20,15     16,22   24,17 18,26   28,19 20,30    32,21   22,34 36,23   24,38




          Standard game-theoretic analysis of the centipede game prescribes that, in

equilibrium, each player selects Down at every decision node. Hence, equilibrium play

prescribes the game ending at the very first node when player A chooses Down. Yet,

study after study finds that subgame perfection fails to organize behavior. As noted in

Levitt et al. (2010), there are a myriad of reasons why subjects may depart from the Nash

strategy and choose not to stop.       It is thus difficult to determine why stopping at the

initial node is such an infrequent occurrence.

          The games outlined below are designed to isolate and measure the impact of four

influences that could lead players to deviate from the equilibrium path. To facilitate such

analysis, we augment the standard centipede game by removing (or enhancing) a

particular confound. Table 1 summarizes the different games used in our experiment.

Each game takes a four-line block with the first block, G1, corresponding to the standard

centipede game.

          The first line of each block contains the game title and the total number of

subjects. The second line lists the strategies, with “A1 D” denoting that player A chooses

Down on the first node, “B1 D” denoting that player B chooses Down on B’s first node,

and so on. The third line in the block contains the payoffs corresponding to the actions

specified at the node. The first number in each pair gives the payoff for player A and the

                                                                                          7
second the payoff for player B. The SPE predictions are in bold and shaded grey. The

fourth line contains the fraction of the relevant subjects choosing each action.

       The first potential confound inherent in the standard centipede game is joint

payoff maximization, i.e., the possibility that subjects play Right instead of Down to

increase the joint payoff to the two players. To isolate the relative influence of such

preferences for efficiency, we augment the standard centipede game in two ways. Game

G2 removes the joint-payoff maximization incentive by holding the combined payoffs

constant at 44 throughout the game. This is a constant sum centipede game of the form

introduced by Fey, McKelvey, and Palfrey (1996) and subsequently tested by and

Bornstein, Kugler, and Ziegelmeyer (2004).

       Note, however, that the SPE payoffs for the constant sum game yield a 50/50 split

of the total surplus and could therefore introduce fairness as a motive for play. Games

G3 through G9 thus take an alternative approach designed to avoid this possible fairness

confounds. These early-move games add pairs of nodes to the beginning of the centipede

game. Playing Right at these added nodes increases not only joint payoffs, but also the

payoffs for both individual players. Hence, SPE play and virtually every behavioral

theory predict that players will choose Right at these early nodes.

       The games G3 through G9 differ along three dimensions; (i) whether they add one

pair or two pairs of nodes before the start of the game, (ii) the growth rates of the payoffs

through the early nodes, and (iii) whether the ensuing game is a standard or a constant

sum centipede. However, in each of these games, the SPE outcome is identical to that

which arises in the corresponding standard (constant sum) centipede game. Similarly, all

payoffs following the SPE node coincide with those that arise in the corresponding game.



                                                                                           8
The second confound is activity bias, which manifests itself in the centipede game

by player A having a desire to play Right in order to create an opportunity for further

participation. In this paper we use an extremely narrow definition of activity bias,

namely that players have a preference for actions that will allow their opponents to have

at least some influence over the outcome of the game and associated payoffs. 7 Games

G10 and G11 address activity bias by giving player B a trivial choice when player A

chooses Down at node A1. For example, consider game G10. If player A chooses Down

at the first node in the otherwise-standard centipede game, player B has the choice

between the payoff combinations (19,10) or (20,15). Game G11 provides a similar

treatment for the constant sum centipede game. Note that the SPE outcomes are identical

to those in games G1 and G2, respectively.

         The third possible confound we consider arises as play may deviate from the

subgame perfect path if rationality is not common knowledge – i.e., as in models built

upon the notion of cognitive hierarchy theory (Camerer et al., 2004) or level-k thinking

(Stahl and Wilson, 1995; Nagel, 1995; Costa-Gomes and Crawford, 2006). Games G12

through G15 are designed to address this possible confound.

         For example, consider game G12, which is obtained by removing the last two

nodes from the standard centipede game and inserting, between nodes B1 and A2, a new

pair of nodes. At these new nodes one player can receive an extremely large payoff and

the other player receives a correspondingly small one, in this case negative. However,

player A can only obtain the really large payoff if player B elects to play Down and


7
 Related versions of activity bias have been noted in a number of contexts such asset market trade (Lei et
al., 2001), penalty kicks in soccer (Bar-Eli et al., 2007), and bargaining games (Carrillo and Palfrey, 2008).
In each of these instances, an aversion to inaction leads players to take potentially costly (suboptimal)
actions.

                                                                                                             9
receive a negative payoff at node B2. Similarly, player B can only obtain the high payoff

if player A opts to take a negative payoff at node A2. Subgame perfection, and virtually

every behavioral hypothesis, predicts that players will choose Right instead of Down at

these nodes.

         Game G14 is very similar to G12 except that the payoffs are lower for both

players at the new nodes than in the previous pair of nodes. In this case player B

choosing Right at node B1 invokes a risk that both players will do worse if A chooses

Down at node A2, in which case playing Right at B1 is less attractive than in the standard

game G1. Games G13 and G15 are similar to G12 and G14 except that they place the

new nodes late in the game rather than early in the game.

         The final confound concerns the lack of focal points in the standard centipede

game. Generally speaking, focal points draw a player’s attention to a subset of his

opponent’s strategy space. In the centipede game a focal point would draw attention to a

player’s action at a particular node. 8 Given that the structure of the centipede game is

fixed, and that the total payoffs grow with each successive node, one way to call attention

to individual nodes is by changing the payoff disparity between the two players. Games

G12 through G17 each insert into the interior of the standard centipede game a pair of

adjacent nodes that are focal by breaking the pattern of payoffs. The focal nodes come

early in games G12, G14, and G16, and late in the game in G13, G15, and G17. 9



8
  For a more detailed discussion on the role of focal strategies in the centipede game, we refer the interested
reader to Neilson and Price (2011) who develop a model of behavioral backward induction who show how
the existence of focal strategies in such games serves to anchor backward induction and increase the
likelihood that players begin the process. The predictions of the model are then tested using a subset of the
data reported in this paper.
9
  As the payoff disparity across nodes is less pronounced in game G14 (G15) than in G12 (G13), we would
expect the focal nodes to be less focal in the former set of games and therefore generate lower rates of SPE
play.

                                                                                                            10
Experimental design

       A total of 202 subjects participated in our laboratory experiment, which was

conducted during the Fall 2009 and Spring 2010 semesters at the University of

Tennessee, Knoxville. Each subject’s experience followed four steps: (1) consideration

of an invitation to participate in an experiment, (2) learning the rules for the centipede

game, (3) actual participation in the centipede game, and (4) conclusion of the

experiment. In step 1, undergraduate students from the University of Tennessee were

recruited using e-mail solicitations.   Once the prerequisite number of subjects had

registered, a second e-mail was sent to each participant confirming their participation in

an experimental session to be held at a given date/time.

       At the start of each session, subjects were seated at linked computer terminals that

were used to transmit all decision and payoff information.          The experiment was

programmed using z-Tree (Fischbacher, 2007). In Step 2, a monitor distributed a set of

instructions after subjects were seated and logged into z-Tree. Subjects were asked to

follow along as the instructions (located in Appendix 1) were read aloud.

        In Step 3, subjects participated in the centipede game. Each session consisted of

12 rounds that lasted about 3 minutes each. At the start of each round, subjects selected

the first node at which they would select Down. Based on these decisions, the computer

determined the final outcome and associated payoffs for each pairing using the process

outlined above. Information on final outcomes and payoffs were displayed on each

subject’s computer screen. Once this information had been displayed for a fixed period

of time (approximately 30 seconds), subjects were shown the game tree for the next

round of play and asked to repeat the decision process.



                                                                                        11
It should be noted that throughout each session careful attention was given to

prohibit communications between subjects that could facilitate cooperative outcomes.

Step 4 concluded the experiment. Subjects completed a post-experiment questionnaire

and were paid their earnings in private.

         Before proceeding, a few key aspects of the experimental design should be

highlighted. First, we randomly assigned each player the role of player A (“White”) or

player B (“Black”) and these roles were maintained throughout all rounds. Second,

subjects were informed that they would be randomly matched with a player of the

opposite type in each of twelve rounds. Importantly, all agents were informed that they

would be matched with a different person in each round and that they would not know the

identity of the person with whom they were matched.

         Third, across sessions we randomized the order in which subjects participated in

each of the twelve games. Fourth, we implemented the strategy method to ensure that we

observed choices for all players in each game. 10

         Fifth, the monitor explained how decisions would be used to determine final

outcomes for each round of play. Once all subjects submitted their final choices, the

computer randomly matched the decisions for each Player A with those for a unique B

Player. Using these decisions, the computer first examined the choice at the initial node

for Player A. If STOP was selected, the game ended. If not, the computer next examined

the decision at the initial node for Player B. If STOP was selected, the game ended. If

not, the computer would examine the decision at the second node for Player A. These

10
   Studies have found that the strategy method and the direct method tend to elicit the same behavior. See
Brandts and Charness (2000), Selten et al. (2003), Oxoby and McLeish (2004), Casari and Cason (2009),
and Fischbacher and Gächter (2009). Moreover, as noted in Brandts and Charness (2010), although the
strategy method tends to induce more selfish play there is no evidence that it impacts treatment effects in a
qualitative sense.

                                                                                                           12
sequential choices continued until the computer reached a node where STOP was selected

or the final node was reached.

       Finally, the monitor explained how final earnings for the experiment would be

determined. After all twelve games were completed, we randomly selected one of the

games by choosing an index card numbered from 1 to 12. The number on the card that

was selected determined which game determined earnings for the session. Subjects were

paid one dollar for every point earned during the selected round. Participants in the

experiment earned an average of $22.90 for a session that lasted about 75 minutes.

3. Testable Hypotheses and Experimental Results

       As noted in Reny (1992), backward induction need not be an optimal strategy in

the centipede game if one were to relax the assumption that maximizing behavior is

common knowledge amongst all players. Moreover, conditioned on player A continuing

at the first node of a centipede chain, it is impossible to consider maximizing behavior

common knowledge throughout the remainder of the game. Exploring behavior beyond

the first node of any centipede chain would thus require a theory of “irrational” behavior

that allows for either non-maximizing behavior and/or relaxes the common knowledge

assumption. This is beyond the scope of the current paper and effectively precludes a

meaningful evaluation of behavior beyond this initial node. As such, we restrict our

analysis to the decision of the A player at the initial node of the centipede chain.

       Table 1 contains the aggregate data from the 17 different games. As noted in the

table, the data show little tendency for SPE play in the standard centipede game. Only

4.0 percent of the player As choose Down at their first nodes. However, subjects do not




                                                                                       13
play Right just once. The average player A selects Right 2.5 times. 11 In this regard, data

for our standard centipede game accord remarkably well with the existing literature.

         As our testable hypotheses concern the frequencies of SPE play, Table 2 collects

these data for all 17 games. It also contains the relevant p-values, derived from the non-

parametric McNemar test, comparing the frequencies in the treatments to the frequencies

in the appropriate baseline games. 12 Figures 2 and 3 show the same frequencies visually,

with the former focusing on games that will be compared to the standard centipede game,

and the latter concentrating on the different variations of the constant sum game.




         Our first testable hypothesis concerns efficiency preferences and a desire for

subjects to maximize joint payoffs. Games G2-G9 are designed to isolate the relative
11
   Of the 17 games we consider, the standard centipede game ranks dead last in all of these categories.
12
   The McNemar test allows the comparison of two population proportions that are correlated to each other.
Our test statistic is thus based on within subject variation in the frequency of subgame perfect play across
games and explicitly controls for the panel nature of our data. All empirical results are robust to the use of
linear probability or related econometric models that explicitly control for factors such as the “round” of
play and allow for correlation across all games and subjects within a given session. Results from these
models are included in a supplemental appendix.

                                                                                                           14
import of such preferences as a driver of play in the standard centipede game. Game G2

removes the joint-payoff maximizing incentive by holding combined payoffs constant at

each node of the game.

       Strategically, the constant sum centipede game is the same as a standard centipede

game absent payoff growth. If players choose Right at their first nodes in the standard

centipede game but Down at their first nodes in the constant sum centipede game, the

behavior would be consistent with efficiency concerns. However, such behavior is far

from conclusive evidence of such preferences. Earlier nodes lead to more equitable

payoff allocations than later ones. An increase in plays of Down at the first nodes of the

constant sum game could thus reflect fairness preferences rather than a response to the

removal of an opportunity to increase joint payoffs.

       Games G3-G9 take an alternate approach that avoids fairness confounds. These

early-move games add pairs of nodes to the beginning of the game for which playing

Right increases the payoffs for both individual players. If play in the standard centipede

game is driven by a desire to increase joint-payoffs, then players should always choose

Right at these added nodes. This leads to our first hypothesis:



Hypothesis 1 (Joint payoff maximization): A larger fraction of subjects play Down at

their first node in the constant sum centipede game G2 than in the standard centipede

game G1. Furthermore, no subjects play Down before the subgame perfect equilibrium

nodes in the early-move games G3 through G9.




                                                                                       15
Hypothesis 1 has two testable implications: (1) SPE play is more frequent in the

constant sum centipede game G2 than in the standard centipede game G1, and (2)

subjects do not play Down too early in the early-move games G3 through G9. As shown

in Table 2, the data from the constant sum centipede game G2 are consistent with the first

part of the hypothesis. The fraction of player As choosing Down at node A1 in this game

increases fourfold from 4.0 percent in G1 to 16.8 percent in G2 – a difference that is

significant at the p < 0.05 level.




        This evidence falls in line with the conventional thinking on behavior in centipede

games, namely that subjects play Right in order to increase their combined payoffs.

However, a deeper understanding of behavior requires looking beyond the conventional

explanation. In the early-move games G3 through G9 subjects who seek to expand joint

payoffs should not choose Down before the SPE nodes. Yet, a non-trivial fraction of all

                                                                                        16
subjects do. Pooled across all early move game, approximately 9.65 percent of all

players and 8.2 percent of player A’s, select Down at a node where playing Right would

have increased payoffs for both players. This is more than double the number of A’s who

play the SPE strategy in the standard centipede game, G1.

        Taken jointly, these data suggest a first result



Result 1. Play in centipede games cannot be organized by the joint payoff maximization

explanation.



Although subjects are significantly more likely to select the SPE nodes when payoffs are

held constant across all choices, a non-trivial fraction of subjects choose Down before the

SPE nodes in our early-move games. Table 3 shows the frequency of playing Down too

early in the seven early-move games. As noted in the table, the incidence of premature

Down plays varies across treatments but is too common to dismiss as noise. 13

        Our second hypothesis concerns activity bias – i.e., an aversion of players to

actions that preclude their opponent from having an influence over the outcome of the

game and associated payoffs. Given our narrow definition, activity bias should not affect

player B as they can only move following an active decision by player A. Consequently,

adding the extra branches at node A1 should have no impact on player B’s decision at

node B1. Observing an increased frequency of choosing Down at node A1 but not at

node B1 would thus illustrate behavior consistent with our definition of activity bias.

This leads to our second testable hypothesis:


13
 Similarly, we observe 10 percent of all players (20 out of 200) selecting down at focal nodes in games
G14 and G15 that yield joint payoffs that are lower than those available at any other node.

                                                                                                          17
Hypothesis 2 (Activity bias): The frequency of subgame perfect equilibrium play for

player A should be greater in game G10 (G11) than in game G1 (G2). There should be

no difference in the frequency of subgame perfect equilibrium play across these games

for player B.



The early-move games G3 through G9 also address activity bias as the early, joint-

payoff-building nodes provide activity for both players. Observing increased frequency

of SPE play in these games compared to the original games (G1 and G2) would be

consistent with an explanation of activity bias.

       Hypothesis 2 outlines two testable implications for play: (1) player A should play

the SPE more frequently in games G10 and G11 than in the corresponding baseline

games, but (2) player B should not. Before discussing the results for these games, it is

important to recall that we employed the strategy method in our experiment. As both

players undertake activities regardless the decision of player A at the first node, our

results likely provide a lower bound on the import of activity bias. Nevertheless, the data

show support for the activity bias explanation.

       For example, consider game G10 which is identical to the standard centipede

game except for giving player B a trivial choice when A plays Down at the first node. As

shown in Table 2, the inclusion of this extra branch in the game tree triples the frequency

of SPE play for player A, from 4.0 percent in the standard game to 11.9 percent in game

G10 – a difference that is significant at the p < 0.05 level. However, there is no

discernable difference in the frequency with which player Bs choose Down at their first



                                                                                        18
node. Hence, behavior in this game fits exactly with the activity bias explanation. The

pattern is less pronounced in game G11, which is based on the constant sum game – there

is no significant difference in the frequency of SPE for either player.

       Taken jointly, these data suggest a second result:



Result 2: Activity bias is a contributing explanation for why subjects fail to play the

subgame perfect equilibrium in centipede games.



Before proceeding, we should note that an alternate test of activity bias comes from a

reconsideration of games G3 through G9, which insert initial moves before the centipede

chain. The activity bias hypothesis predicts greater SPE play for player A but not for

player B – a prediction borne out in our data. As noted in Table 2, the frequency of SPE

play for A players in these games is significantly greater than that observed in the

corresponding standard (constant sum) baseline. Our “early move” games therefore

provide additional support for Hypothesis 3. Changing the game to allow moves for

player B leads to increased SPE play for player A.

       Hypothesis 3 concerns players’ beliefs about the rationality of their opponents.

Games G12 through G15 were designed specifically to test this hypothesis. For example,

consider game G12. Suppose that, in the standard centipede game G1, player A’s beliefs

about B’s behavior at node B1 make it a best response to play Right at node A1.

Assuming that beliefs are consistent across games, it should now be even more attractive

to play Right at node A1. A is no worse off if B plays Right at B1, but could be much

better off if B were to subsequently play Down at B2. So, for any beliefs that make A



                                                                                     19
play Right at A1 in the standard centipede game, those same beliefs should make A play

Right at A1 in game G12.

       Game G14 is similar to G12 except the payoffs for both players at the new nodes

are lower than those available in the preceding node. In this case player A choosing

Right at node A1 invokes a risk that both players will do worse if B chooses Down at

node B2. Hence, playing Right at A1 is less attractive than in the standard game G1.

This leads to our third testable hypothesis:



Hypothesis 3 (Responses to beliefs about opponent behavior): Compared to game

G1, subjects should play Right at their first nodes with greater frequency in games G12

and G13, and Down at their first nodes with greater frequency in games G14 and G15.



       The data in Table 2 make it obvious that Hypothesis 3 fails.      The frequency of

subgame perfect play in these new treatments is higher than that observed in the

corresponding baseline games. For example, consider Game G12 which is based on the

standard centipede game. The frequency of choosing Down at their first nodes increases

tenfold to 41.2 percent for player A – a difference that is statistically significant at

conventional levels. Similar patterns hold in game G13, with increased frequencies of

subgame perfect play rather than the hypothesized reduced frequencies.

       Hypothesis 3 prescribes the opposite pattern for games G14 and G15 – there

should be higher frequencies of playing Down at the first nodes in these games than in

G1. The data support this prediction. For example, the frequency of subgame perfect

play in Game G14 increases six-fold to 24 percent for player A – a difference that is



                                                                                      20
significant at conventional levels.     We observe similar effects in game G15 – the

frequency of SPE play increase by 18 percentage points.

       Yet, one must question whether this really drives behavior when games G12 and

G13 contradicted the hypothesis so readily and have higher frequencies of subgame

perfect play. Given this, our data suggest a third result.



Result 3: Play in centipede games cannot be organized by players best-responding to

their beliefs about their opponents’ rationality.



A direct implication of Result 3 is that there are few, if any, level-1 thinkers in our

subject pool. Level-1 thinkers best respond to random play by their opponent. Random

opponent play in game G12 makes playing Right at the first node much more attractive

than in the standard centipede game.

       To see this, consider the decision facing Player A in this game. Playing Right at

node A1 in the standard centipede game yields a 50:50 chance of earning 16 when player

B plays Down at node B1 or earning 24 when A plays Down at node A2. In Game G12

playing Right at node A1 yields a 50 percent chance of earning 16 when B plays Down at

node B1, or, after B plays Right at node B1, a 25 percent chance of earning 45 when

player B plays Down at node B2 or a 25 percent chance of earning 24 when A plays

Down at node A3. Player A’s payoff distribution from playing Right in game G12 first-

order stochastically dominates that from playing Right in the standard centipede game.

Hence, any level-1 thinker should select Right at node A1 of this game. Yet, we do not




                                                                                     21
observe any A players selecting the SPE strategy in the standard game but Right at node

A1 in game G12.

         Similarly, when combined with data from our “early-move” games (G3-G9),

results from Game G12 and G13 are at odds with the predictions of Jehiel’s (2005)

Analogy-Based Expectation Equilibrium. Under Jehiel’s model, the inclusion of nodes

for which the play of Right is an obvious choice – as with nodes A1 and B1 in all early-

move games – should lead to an increase in the likelihood subjects’ select Right at later

nodes. Intuitively, the inclusion of such nodes would “bias” upwards aggregate pass

rates and, based on this information, make Right a more “attractive” option at every

subsequent node. Yet, our data suggest that the inclusion of such nodes lowers the

likelihood of playing Right at the SPE and all subsequent nodes.

         The final hypothesis concerns the lack of focal points in the standard centipede

game. Games G12 through G17 introduce focal points by breaking the interior payoff

structure of the game. 14 This should simplify the solution of the game and thus facilitate

SPE play. Intuitively, focal points draw attention to a particular element of an opponent’s

strategy space. In doing so, focal nodes help anchor backward induction and increase the

likelihood of starting the process (Neilson and Price, 2011).

         This gives rise to our final hypothesis:




14
   There are many ways that one can make certain strategies or nodes focal. One could draw attention
through the use of visual cues like colors or labels as has been done in coordination games (see, e.g., Mehta
et al., 2994; Crawford et al., 2008). However, such changes would reflect properties of the presentation of
the game rather than the game itself. We have chosen an alternate approach and introduce “focalness”
through changes to the properties of the game itself. However, the change we consider are innocuous in the
sense that they have no impact on equilibrium play.

                                                                                                          22
Hypothesis 4 (Focal points): The fraction of subjects playing Down at their first node

should be higher in games G12, G14, and G16 than in games G13, G15, and G17, and

these in turn should be higher than the fraction in game G1.



It is important to note that Hypotheses 3 and 4 are contradictory. Hypothesis 3 states that

subjects should play Right more frequently in games in which Hypothesis 4 says they

should play Down more frequently. Thus, not only do games G12 through G15 provide a

test between subgame perfection and best responses to beliefs about opponent

irrationality, they provide a direct test between the latter and backward induction using a

focal point.

        The most striking evidence regarding focal points is the introduction of a focal

node in game G16. This game maintains from the standard centipede game both the

growth of the total payoffs and the identity of who gets the majority share, but changes

the size of the payoff disparity in the second pair of nodes – the deciding player gets the

entire payoff and the other player a payoff of zero. Although this change to the game

makes no difference strategically, it increases the frequency of subgame perfect behavior

by a factor of eight. The frequency of subgame perfect behavior for Player A increase

dramatically – going from 4.0 percent in the standard centipede game to 34.7 in this new

game.

        Game G17 makes a similar payoff change but to the fourth pair of nodes instead

of the second pair of nodes. Once again the frequency of SPE play increases to 19.8

percent for player A. The decline in the rate of SPE play here actually bolsters the

explanation of focal points. If players backward induct from the focal point, they face an



                                                                                        23
easier task in game G16 and a more difficult one in G17 as they must induct more steps.15

Moreover, the observed data patterns are consistent with results from McKelvey and

Pelfrey (1992) who find a nearly 10-fold increase in SPE play when moving from a six-

to a four-move centipede game.

        Games G12 and G13 also have pairs of focal nodes provided by the negative

payoffs to one of the players. Since the player potentially earning these negative payoffs

is the one making the decision at that node, these negative payoffs are easily avoided and

do not matter. However, if players use the focal nodes as an anchor for backward

induction, we would expect increased SPE play in these games. Empirical evidence

supports this prediction. Approximately 41.2 percent of the A players select Down at

their first nodes in game G12 and approximately 13.7 percent of such players choose

Down at their first nodes in game G13.

        By avoiding zero or negative payoffs, games G14 and G15 make these nodes less

focal. Structurally they are the same as games G12 and G13 with the player making the

decision at the node with the really low payoff. If focal points are driving the behavior,

and negative payoffs (or payoffs with a greater disparity) are viewed as more focal, one

would expect less SPE play in game G14 than in game G12 and less in game G15 than in

game G13. The evidence bears out part of this prediction. For player A, playing Down

at the first node is much more common in game G12 than in G14. However, the pattern

reverses for the games with late focal points.

        Taken jointly, these data suggest a final result.


15
  That the difficulty of a task influences the frequency of SPE play in our setting, shares similarity with
results from Ho and Weigelt (1996) who show that the complexity of a decision task influences equilibrium
selection in coordination games. This is also consonant with Neilson and Price (2011) who show that the
length of a centipede chain is inversely related to the expected frequency of SPE play.

                                                                                                        24
Result 4: The lack of focal points is a contributing explanation for why subjects fail to

play the subgame perfect equilibrium in centipede games.



In fact, with the exception of the constant sum game G2, all of the variants of the

standard centipede game shown in Figure 2 generate focal points by breaking the pattern

of payoffs somewhere. And, interestingly, all of them have higher frequencies of SPE

play.

        In the activity bias game G10 this higher frequency is unlikely to arise from

backward induction from a focal node because the change to the game occurred in the

very first node making further backward induction impossible.             Thus, the evidence

suggesting a role for activity bias still stands in the presence of focal points, but the key

finding here is that focal points play a role in anchoring the backward induction process.

Furthermore, final nodes do not seem to be focal in the same way that interior nodes do,

possibly because it is difficult to detect a break in a pattern at an endpoint.

4. Conclusions

        By making small changes to the payoff structure, but not the strategy space or the

equilibrium path, of the standard centipede game our experiments provide a horse race

over different explanations of why subjects fail to play the subgame perfect equilibrium.

As with any horse race, our data suggest that there are winners and losers. The biggest

winner is the concept of focal points. As normally presented the centipede game has no

focal points because every node involves the same tradeoffs. Our experiment adds focal

nodes that break the pattern of payoffs. Regardless of whether players should play Down



                                                                                          25
or Right at such nodes, our results have increased frequencies of SPE play. Importantly,

this suggests that players are drawn to the focal nodes and this facilitates the backward

induction process.

       We also find evidence supporting a particularly strong form of activity bias – i.e.,

subjects have a preference for wanting opponents to have an influence of outcomes of the

game and associated payoffs. Such a finding is surprising as the experiments used the

strategy method. Hence, the first and second movers in our experiment make exactly the

same number of choices.      Still, the evidence for activity bias persists, and is not

subsumed by the evidence for focal points.

       There are also losers. Explanations based on the notions that (i) subjects possess

a desire to maximize joint payoffs or (ii) playing Right is a best response to subjects’

beliefs about their opponents’ irrationality fail to stand up to the evidence. Yet, we

would be remiss to suggest that these considerations have no part in behavioral game

theory. Instead, they simply suggest that these explanations may not be important forces

driving how subjects play in the centipede game.




                                                                                        26
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                                                                                   29
Table 1

                                            G1: Standard centipede (n=202)
A1 D     B1 D     A2 D     B2 D          A3 D       B3 D      A4 D      B4 D           A5 D     B5 D     A5 R    B5 R
20, 15   16, 22   24, 17   18, 26       28, 19     20, 30     32, 21     22, 34        36, 23   24, 38    --     40, 25
4.0%     8.9%     26.7%    27.7%         26.7%     24.8%      14.9%      19.8%         16.8%    10.9%    10.9%   7.9%
                                          G2: Constant sum centipede (n=202)
 A1 D     B1 D     A2 D     B2 D         A3 D       B3 D       A4 D      B4 D          A5 D     B5 D     A5 R    B5 R
22, 22   20, 24   26, 18   15, 29       31, 13     11, 33     34, 10     7, 37         40, 4    2, 42     --     44, 0
16.8%    38.6%    47.5%    39.6%         20.8%     11.9%       6.9%      7.9%          5.9%     2.0%     2.0%    0.0%
                                G3: One-sided error one move standard 1 (n=102)
A1 D     B1 D     A2 D     B2 D      A3 D       B3 D      A4 D       B4 D       A5 D            B5 D     A5 R    B5 R
12, 9    17, 14   20, 15   16, 22       24, 17     18, 26     28, 19     20, 30        32, 21   22, 34    --     36, 23
0.0%     0.0%     19.6%    35.3%         35.3%     25.5%      17.6%      21.6%         23.5%    11.8%    3.9%    5.9%
                                    G4: One-sided error two moves standard 1 (n=102)
A1 D     B1 D     A2 D     B2 D          A3 D       B3 D       A4 D       B4 D         A5 D     B5 D     A5 R    B5 R
 8, 5    10, 7    12, 9    17, 14       20, 15     16, 22     24, 17     18, 26        32, 21   22, 34    --     36, 23
0.0%     0.0%     0.0%     5.9%         35.3%      33.3%      33.3%      23.5%         19.6%    33.3%    11.8%   3.9%
                                G5: One-sided error one move standard 2 (n=100)
A1 D     B1 D     A2 D     B2 D      A3 D       B3 D      A4 D       B4 D       A5 D            B5 D     A5 R    B5 R
 8, 5    10, 7    20, 15   16, 22       24, 17     18, 26     28, 19     20, 30        32, 21   22, 34    --     36, 23
14.0%    14.0%    14.0%    38.0%         32.0%     14.0%      22.0%      22.0%         6.0%     8.0%     12.0%   4.0%
                                    G6: One-sided error two moves standard 2 (n=100)
A1 D     B1 D     A2 D     B2 D          A3 D       B3 D       A4 D       B4 D         A5 D     B5 D     A5 R    B5 R
12, 9    14, 10   15, 12   17, 14       20, 15     16, 22     24, 17     18, 26        32, 21   22, 34    --     36, 23
2.0%     8.0%     14.0%    4.0%         32.0%      38.0%      12.0%      30.0%         20.0%    12.0%    20.0%   8.0%




                                                                                                                          30
G7: One-sided error one move constant sum 1 (n=102)
 A1 D      B1 D     A2 D     B2 D     A3 D        B3 D      A4 D        B4 D      A5 D     B5 D     A5 R    B5 R
 20, 19    22, 21   22, 22   20, 24    26, 18     15, 29      31, 13     11, 33   34, 10   7, 37     --     40, 4
 2.0%      11.8%    31.4%    45.1%     39.2%      29.4%       9.8%        5.9%    11.8%    3.9%     5.9%    3.9%
                                 G8: One-sided error one move constant sum 2 (n=100)
 A1 D      B1 D     A2 D     B2 D       A3 D       B3 D       A4 D        B4 D     A5 D    B5 D     A5 R    B5 R
 16, 15    22, 21   22, 22   20, 24    26, 18     15, 29      31, 13     11, 33   34, 10   7, 37     --     40, 4
 14.0%     8.0%     28.0%    62.0%     40.0%      18.0%       2.0%        4.0%    4.0%     6.0%     12.0%   2.0%
                                 G9: One-sided error two moves constant sum (n=202)
 A1 D      B1 D     A2 D     B2 D      A3 D        B3 D      A4 D       B4 D      A5 D     B5 D     A5 R    B5 R
 16, 15    18, 17   20, 19   22, 21    22, 22     20, 24      26, 18     15, 29   31, 13   11, 33    --     34, 10
 0.0%      4.0%     8.9%     11.9%     35.6%      61.4%       45.5%      16.8%    8.9%     5.0%     1.0%    1.0%
                                          G10: Activity bias standard (n=202)
 A1 D      B1 D     A2 D     B2 D       A3 D       B3 D       A4 D        B4 D     A5 D    B5 D     A5 R    B5 R

   B
chooses
19,10 or
 20, 15    16, 22   24, 17   18, 26    28, 19     20, 30      32, 21     22, 34   36, 23   24, 38    --     40, 25
 11.9%     13.9%    24.8%    23.8%     20.8%      23.8%       13.9%      14.9%    21.8%    21.8%    6.9%    2.0%
                                       G11: Activity bias constant sum (n=202)
  A1 D     B1 D     A2 D     B2 D      A3 D        B3 D        A4 D       B4 D     A5 D    B5 D     A5 R    B5 R
   B
chooses
 21, 20
 or 22,
   22      20, 24   26, 18   15, 29    31, 13     11, 33      34, 10      7, 37   40, 4    2, 42     --     44, 0
 21.8%     38.6%    41.6%    32.7%     13.9%      10.9%       5.9%        9.9%    9.9%     5.0%     6.9%    3.0%




                                                                                                                     31
G12: Early beliefs 1 (n=102)
A1 D     B1 D     A2 D     B2 D     A3 D      B3 D         A4 D       B4 D          A5 D     B5 D     A5 R    B5 R
20, 15   16, 22   -5, 44   45, -5   24, 17       18, 26      28, 19        20, 30   32, 21   22, 34    --     36, 23
41.2%    41.2%    0.0%     5.9%     25.5%        17.6%       7.8%          13.7%    11.8%    11.8%    13.7%   9.8%
                                             G13: Late beliefs 1 (n=102)
A1 D     B1 D     A2 D     B2 D     A3 D          B3 D       A4 D          B4 D     A5 D     B5 D     A5 R    B5 R
20, 15   16, 22   24, 17   18, 26   28, 19       20, 30      -5, 56        57, -5   32, 21   22, 34    --     36, 23
13.7%    19.6%    31.4%    37.3%    41.2%        27.5%       0.0%          5.9%     7.8%     9.8%     5.9%    0.0%
                                         G14: Early beliefs 2 (n=100)
A1 D     B1 D     A2 D     B2 D     A3 D      B3 D         A4 D       B4 D          A5 D     B5 D     A5 R    B5 R
20, 15   16, 22   12, 9    17, 4    24, 17       18, 26      28, 19        20, 30   32, 21   22, 34    --     36, 23
24.0%    22.0%    16.0%    8.0%     10.0%        24.0%       14.0%         16.0%    24.0%    22.0%    12.0%   8.0%
                                    G15: Late beliefs 2 (n=100)
A1 D     B1 D     A2 D     B2 D     A3 D          B3 D       A4 D          B4 D     A5 D     B5 D     A5 R    B5 R
20, 15   16, 22   24, 17   18, 26   28, 19       20, 30      12, 9         17, 14   32, 21   22, 34    --     36, 23
22.0%    26.0%    24.0%    38.0%    26.0%        20.0%       6.0%          10.0%    8.0%     6.0%     14.0%   0.0%
                                         G16: Early focal point (n=202)
A1 D     B1 D     A2 D     B2 D     A3 D          B3 D       A4 D          B4 D     A5 D     B5 D     A5 R    B5 R
20, 15   16, 22   39, 0    0, 40    24, 17       18, 26      28, 19        20, 30   32, 21   22, 34    --     36, 23
34.7%    70.3%    46.5%    9.9%     5.0%          5.0%       3.0%          4.0%     4.0%     3.0%     6.9%    7.9%
                                           G17: Late focal point (n=202)
A1 D     B1 D     A2 D     B2 D     A3 D          B3 D       A4 D          B4 D     A5 D     B5 D     A5 R    B5 R
20, 15   16, 22   24, 17   18, 26   28, 19       20, 30      51, 0         0, 52    32, 21   22, 34    --     36, 23
19.8%    17.8%    14.9%    27.7%    23.8%        38.6%       31.7%         7.9%     4.0%     5.0%     5.9%    3.0%




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