Activity Bias and Focal Points in the Centipede Game →
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. firstname.lastname@example.org. Neilson: Department of Economics, University of Tennessee, Knoxville. email@example.com. Price: Department of Economics, University of Tennessee, Knoxville and NBER. firstname.lastname@example.org. 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.
1 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 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.
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
2 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 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. 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. 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.
3 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.
4 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 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.
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. 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 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.
To examine whether the failure to fully backward induct in the centipede game is driven by a lack of focal points, we 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).
5 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
6 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.
7 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
8 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.
9 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 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. 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.
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.
10 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.
11 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.
12 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.
13 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
14 play Right just once. The average player A selects Right 2.5 times.11 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. In this regard, data for our standard centipede game accord remarkably well with the existing literature. 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.
15 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.
16 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
17 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.
18 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
19 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
20 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
21 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
22 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 gives rise to our final hypothesis: 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).
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.
23 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
24 easier task in game G16 and a more difficult one in G17 as they must induct more steps.15 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.
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. 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.
25 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
26 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.
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30 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%
31 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%
32 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%
33 Table 2 Frequencies of subgame perfect equilibrium play Game Player A Player B Frequency p-value against std. centipede p-value against constant sum Frequency p-value against std. centipede p-value against constant sum G1 0.040 0.089 G2 0.168 0.006 0.386
34 Table 3 Fraction of subjects playing Down too soon Player A Player B G3 0.00 0.00 G4 0.00 0.06 G5 0.14 0.14 G6 0.16 0.12 G7 0.02 0.12 G8 0.14 0.08 G9 0.09 0.16
35 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.
36 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.