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. [email protected]. Neilson:
Department of Economics, University of Tennessee, Knoxville. [email protected]. Price: Department of
Economics, University of Tennessee, Knoxville and NBER. [email protected].
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 (BoschDomenech 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
and
πi,t + 1 < πi,t < πi,t + 2
(1)
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 nonparametric 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.
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.
12
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, jointpayoff-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 firstorder 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 earlymove 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 sixto 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)
A3 D
B3 D
A4 D
B4 D
A5 D
B5 D
A5 R
B5 R
A1 D
B1 D
A2 D
B2 D
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
22, 22
B1 D
20, 24
A2 D
26, 18
B2 D
15, 29
A3 D
31, 13
B3 D
11, 33
A4 D
34, 10
B4 D
7, 37
A5 D
40, 4
B5 D
2, 42
A5 R
--
B5 R
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)
B2 D
A3 D
B3 D
A4 D
B4 D
A5 D
B5 D
A5 R
B5 R
A1 D
B1 D
A2 D
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%
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)
B2 D
A3 D
B3 D
A4 D
B4 D
A5 D
B5 D
A5 R
B5 R
G4: One-sided error two moves standard 1 (n=102)
A1 D
B1 D
A2 D
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)
B2 D
A3 D
B3 D
A4 D
B4 D
A5 D
A1 D
B1 D
A2 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)
B2 D
A3 D
B3 D
A4 D
B4 D
A5 D
B5 D
A5 R
B5 R
A1 D
B1 D
A2 D
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%
A1 D
B
chooses
21, 20
or 22,
22
B1 D
A2 D
B2 D
G11: Activity bias constant sum (n=202)
A3 D
B3 D
A4 D
B4 D
A5 D
B5 D
A5 R
B5 R
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
A1 D
B1 D
A2 D
B2 D
G12: Early beliefs 1 (n=102)
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%
A5 D
B5 D
A5 R
B5 R
A1 D
B1 D
A2 D
B2 D
G14: Early beliefs 2 (n=100)
A3 D
B3 D
A4 D
B4 D
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%
32
Table 2
Frequencies of subgame perfect equilibrium play
Player A
Game
G1
G2
G3
G4
G5
G6
G7
G8
G9
G10
G11
G12
G13
G14
G15
G16
G17
Frequency
0.040
0.168
0.196
0.353
0.140
0.320
0.314
0.280
0.356
0.119
0.208
0.412
0.137
0.240
0.220
0.347
0.198
p-value against
std. centipede
0.006
0.013
0.0002
0.182
0.001
0.043
0.0001
0.131
0.008
0.027
0.0001
0.002
Player B
p-value against
constant sum
0.149
0.146
0.001
0.540
Frequency
0.089
0.386
0.353
0.333
0.380
0.380
0.451
0.620
0.614
0.139
0.386
0.412
0.196
0.220
0.260
0.703
0.178
p-value against
std. centipede
<0.0001
0.022
0.016
0.0005
0.0002
0.267
0.0007
0.343
0.027
0.009
<0.0001
0.067
p-value against
constant sum
0.689
0.019
0.001
0.874
33
Table 3
Fraction of subjects playing Down too soon
G3
G4
G5
G6
G7
G8
G9
Player A
0.00
0.00
0.14
0.16
0.02
0.14
0.09
Player B
0.00
0.06
0.14
0.12
0.12
0.08
0.16
34
Appendix 1:
INSTRUCTIONS
Thank you for participating in this experiment on decision-making behavior. You will be paid for your
participation in cash at the end of the experiment. Your earnings for today’s experiment will depend
partly on your decisions and partly on the decisions of the player with whom you are matched.
It is important that you strictly follow the rules of this experiment. If you disobey the rules, you will be
asked to leave the experiment.
If you have a question at any time during the experiment, please raise your hand and a monitor will come
over to your desk and answer it in private.
Description of the task
You will be participating in a simple game. The game requires 2 players, one of whom will be called
Player A and the other Player B. Prior to the start of the session, you will be randomly assigned the role
of either Player A or Player B and will remain in this role throughout the experiment.
Each player has to choose between two decisions:
STOP
or
CONTINUE
for each of 5 decision nodes. As soon as any player chooses to STOP, the game ends. If a player chooses
to CONTINUE, the other player will be faced with the same choice: STOP or CONTINUE. If he is the
last player in the sequence, the game will end regardless of what decision he makes.
Player A will make the first decision. As indicated above, the game ends as soon as one player chooses to
STOP. Below is a pictorial representation of the game. The color of the circles (WHITE or BLACK)
identifies which player makes a decision (either STOP or CONTINUE) given that the game has
progressed to that circle. The arrows pointing right and down represent the two decisions. The terminal
brackets contain the payoff information. The game will end at one of the eleven terminal brackets.
All of the payoffs are in U.S. dollars. The top number in each bracket identifies the payoff in $’s for
Player A. The bottom number in each bracket indentifies the payoff in $’s for Player B.
The game will start with Player A at the farthest left decision node. Please take some time now to study
the structure of the game.
35
The experiment consists of 12 games. In each game you are matched with a different player of the
opposite type. That is, if you are Player A you will be matched with a different Player B for each
subsequent game. Importantly, you will not know the identity of the players with whom you will be
matched, nor will the person with whom you are matched know your identity.
Procedure for Playing the Game:
Indicate on your computer screen at which node you would first like to choose STOP by pressing the
button that corresponds to that particular node. If you wish to play continue for all five of your nodes,
please press the None option. Once you have made your selection, please press the submit button to
record your final decision.
Once all subjects have made their decisions, the computer will randomly match the decisions for each
Player A with the decision for a unique B Player.
Using the decisions for each player, the game will be played out as follows. The computer will examine
the decision at the first node for Player A. If he selected STOP for this node, the game will end. If not,
the computer will examine the decision at the first node for Player B. Again, if he selected STOP for this
node, the game will end. If not, the computer will examine the decision at the second node for Player A.
These sequential choices continue until we reach either a node where STOP was selected or the final node
– the one farthest right – is reached.
Once the outcome of the game has been determined by the computer, you will be informed of the
outcome of the game (the node at which STOP was first selected) along with the associated payoff.
This same basic procedure will be followed for each of twelve games.
Determining Final Payoffs
You will only be paid your earnings for one of the twelve games you will play during today’s session.
After all twelve games have been completed, we will randomly select one of the games by selecting an
index card that is numbered from 1 to 12. The number on the card which is selected will determine which
game will determine your earnings for today’s session.
Even though you will make twelve decisions, only one of these will end up affecting your earnings. You
will not know in advance which decision will hold, but each decision has an equal chance of being
selected.
36