On the relevance of irrelevant strategies

Abstract

The experimental literature on individual choice has repeatedly documented how seemingly-irrelevant options systematically shift decision-makers’ choices. However, little is known about such effects in strategic interactions. We experimentally examine whether adding seemingly-irrelevant strategies, such as a dominated strategy or a duplicate of an existing strategy, affects players’ behavior in simultaneous games. In coordination games, we find that adding a dominated strategy increases the likelihood that players choose the strategy which dominates it, and duplicating a strategy increases its choice share; The players’ opponents seem to internalize this behavior and best respond to it. In single-equilibrium games, these effects disappear. Consequently, we suggest that irrelevant strategies affect behavior only when they serve a strategic purpose. We discuss different theoretical approaches that accommodate the effect of salience and may explain our findings.

Appendices

Appendix A: details of experimental design

1.1 Appendix A1: payoff matrices

For robustness purposes, for each game type we examined four different payoff matrices that slightly varied in their monetary payoffs and in the location of equilibria. The construction of the base games and their extensions followed a set of predetermined rules. Below we describe the main rules alongside a brief explanation of their underlying rationale. The payoff matrices appear in Tables 19 and 20.

• 1. Coordination base games are symmetric which allows for a swift understanding of the base game. The equilibrium payoffs on the other hand are asymmetric, i.e., (x, y) and (y, x) where $x \ne y.$

• 2. In the extended games, the added strategy generates the same payoff to the column player regardless of his own action. This reduces the potential for direct effects on the column players’ behavior so that any effect on the column players is more likely to be a reaction to the expected change in the behavior of the row players due to the added strategy.

• 3. In the dominance extensions, the last digit of the row player’s payoffs in the dominated strategy is different than the last digit of the other payoffs. In addition, the column player’s payoff when the row player chooses the dominated option is 10 ILS lower than his lowest payoff in the base game. These features emphasize the domination relation and increase the likelihood that subjects will notice it.

• 4. In the compromise extensions, when the row player chooses the added strategy, the column player’s payoff is equal to his lowest payoff in the base game.¹⁶

• 5. Payoffs are multiplications of 5 for clarity and simplicity.

Table 19 Payoffs of coordination base games alongside their extensions

Table 20 Payoffs of single-equilibrium base games alongside their extensions

1.2 Appendix A2: Order of play

Table 21

Table 21 Order of games in Version 1 for players in Group 1

1.3 Appendix A3: Instructions

Welcome to the experiment

You are about to participate in an interactive decision making experiment. Please follow the instructions carefully.

In the experiment you may earn a significant amount of money. For your participation you will receive 20 ILS. You may earn an additional substantial amount based on your decisions and the decisions of the other participants in this room.

During the experiment you will play 36 games. In each game you will be randomly matched with another participant as the opponent against whom you will play the game. The game will be presented on your screen and the interaction between you and your opponent will take place through the computer. The identity of your opponents will not be revealed to you during the experiment or after it is completed. In every game you may earn different sums of money depending on your choice and the choice of your opponent. Upon completion of the experiment, the computer will randomly draw one of the 36 games you played and the amount of money that you earned in that game will be paid to you. Each participant may have a different game chosen for payment. The choices of your opponent and payoffs will not be presented during the experiment but only upon its completion. Upon completion, you will learn your payoff in each game and which game was chosen for payoff.

Note that since nobody (not even the experimenters) know which game will be chosen for payment purposes, it is best for you to treat every game as if it is the one that counts. The total amount of earnings in the experiment (participation fee and the amount earned in the randomly drawn game) will be paid to you privately in cash immediately after the experiment is completed. We will move on to the payment stage only after all participants finish marking their choices in all games.

It is not allowed to talk during the experiment or to look at other participants’ screens. If you have any questions please raise your hand and one of the experimenters will be happy to answer. In most games you will see a table of the following type:

	Left	Right
Up	50, 40	10, 20
Down	70, 20	30, 60

One of the participants will be considered the row player and the other participant will be considered the column player. In the game’s instructions it will be mentioned if you are playing as the row player or the column player in that game.

The actions described in the rows are the actions that the row player can choose from. In the above table, these are Up and Down.

The actions described in the columns are the actions that the column player can choose from. In the above table, these are Left and Right.

Each player will be asked choose an action without knowing the other player’s chosen action. In games in which you are the row player, another participant sees the same table and plays against you as the column player. When you are playing the role of the column player, another participant is playing against you as the row player.

The numbers in the cells of the table represent the ILS amount that each one of you will receive for any combination of your choices. In each cell, the payoff for the row player always appears on the left and the payoff to the column player always appears on the right.

For example, if the row player chose Up and the column player chose Left then the row player will receive a payoff of 50 ILS and the column player will receive a payoff of 40 ILS. If the row player chose Down and the column player chose Right then the row player will receive a payoff of 30 ILS and the column player will receive a payoff of 60 ILS.

In some games you will play the role of the row player and in some games you play the role of the column player.

In any game that you will play, regardless of your role, your payoffs will always be in blue while the payoffs of the other player will be in black. The purpose of the colors is simply to assist you in recognizing your own payoffs. Remember the rule: Blue is mine, Black is the opponent’s.

A few games in the experiment will be described verbally and will not include a payoff table.

5 Training Games

In the first part of the experiment, you will play 5 training games to make sure that you understand the instructions. You will not receive payoffs for your choices in this training session. Following the training session, you will move on to the 36 games in which you may earn payoffs.

Appendix B: Additional results

1.1 Appendix B1: Dominated strategy effects including all observations

In this section, we reproduce the dominance-extension analysis when we do not exclude the choices of the decoy strategy. Table 22 reproduces Table 3 from Sect. 4 and shows similar patterns of behavior: The absolute percentages of choices of the target strategy moderately increase in all coordination extended games (2–10%) and weakly increase in single-equilibrium extended games (0–6%). Table 23 reports the regressions with the dominance extension dummy variable for coordination games (specifications (1)–(3)) and single-equilibrium games (specifications (4)–(6)).¹⁷ The results are qualitatively similar to those reported in the main text, albeit the coefficient on the dominance extension variable is of lower significance.¹⁸

Table 22 Percentages of target choices by row players (all observations)

Table 23 Logistic regression models: dominance extension with all observations

1.2 Appendix B2: Accounting for the effect of experience

In this section we examine whether the effects of the added strategies that were reported in the main text vary with subjects’ experience. Although subjects did not receive any feedback on the outcome of play after each game, experience may affect subjects’ behavior. For example, it is possible that it takes time to understand the underlying structure of the games and the potential gains that may arise by following the behavioral cues in the extended games. To explore this possibility, we leverage a feature of our experimental design–subjects were randomly assigned to one of two versions with two opposite orders of the 32 games. Thus, the set of the first 16 games for one group of subjects is identical to the set of the last 16 games for another group of subjects. We define a dummy variable, Early, that receives 1 if the game appeared in the first 16 games that the subject encountered and 0 otherwise. We rerun the main regressions that appeared in the main text but this time we add Early as an explanatory variable, as well as an interaction between Early and Extension (i.e., the dummy for the extended game). Our main interest in this section lies in the coefficient of the interaction variable which captures the difference in the effect of the extensions across early and late games. Our findings are reported in the six tables below. The first two tables describe row players’ choices in coordination games and single-equilibrium games, respectively. These are followed by the regressions for the column players’ choices. Finally, we report the corresponding regressions of the compromise extensions.

The tables show that in most instances, the coefficient of the interaction variable is not significantly different from zero. In other words, the effect of the extensions was relatively similar across early and late games. There are three instances (out of 12) in which the coefficient of the interaction variable is significant at the 5% or the 10% level. For example, Table 24 suggests that the dominance extension has a stronger effect on row players’ choices in the later stages of the experiment compared to the earlier stages.

Overall, we conclude that experience did not play a crucial role in our experiment; In most games the impact of extensions does not significantly differ between early and late stages. In the instances in which such a difference does show up, the behavior in the later games is the one that sets the tone for the overall effect that showed up in our main analysis

Table 24 Logistic regression models: row players’ choices in coordination games

Table 25 Logistic regression models: row players’ choices in single-equilibrium games

Table 26 Logistic regression models: column players’ choices in coordination games

Table 27 Logistic regression models: column players’ choices in single-equilibrium games

Table 28 Logistic Regression Models: Compromise Extension in Coordination Games

Table 29 Logistic regression models: compromise extension in single-equilibrium games

Appendix C: Theoretical models

1.1 Appendix C1: The generalized cognitive hierarchy model

1.1.1 Irrelevant strategies and the generalized cognitive hierarchy model

In this section, we make use of the Generalized Cognitive Hierarchy (GCH) Model (Chong et al., 2016), with one slight adjustment, to shed light on our findings. We think of this exercise as a formal illustration of one potential channel through which coordination may increase in the presence of irrelevant strategies. Throughout this section, we focus on the qualitative difference in the model’s predictions between the base games and their extensions.

The GCH model is a generalization of Cognitive Hierarchy (CH) theory (Camerer et al., 2004). In CH, level-k players, for $k \ge 1$, do not best respond to level-$(k-1)$ players, as in the standard level-k model, but rather to the population of lower-level players whose types are drawn from a Poisson distribution; level-0 players choose each action with equal probability. GCH generalizes this model in two respects. First, it allows players to use “stereotypes," i.e., assign a disproportional higher weight to frequently occurring lower-level types. Second, it modifies the behavior of level-0 players: While in the standard level-k model, they choose each strategy with equal probability, in GCH they are more likely to choose from a set of strategies that never yield the minimal payoff given any strategy of the opponent (which is dubbed the “never worst set" of strategies). If this set is empty, then they choose randomly with equal probabilities as in CH and the standard level-k model.

1.1.1.1 Coordination games

According to the GCH model, level-0 row players are more likely to choose the target in the dominance extensions, where it belongs to the never worst set of strategies, than in the corresponding base games. They also increase their choice probability of the duplicated strategy but for a different reason: Since there are no strategies that are never worst, each strategy is played with equal probability. As a result, the duplicated strategy is chosen by level-0 players with a probability of 2/3 (compared to 1/2 in the base game). Level-0 column players behave the same across base games and extensions since their never-worst set is not affected by the addition of the dominated and duplicated strategies.

Let us now move to the next level of cognitive hierarchy. We start with level-1 column players who best respond to level-0 row players. In the base games, their action depends on their level of risk aversion. If they are risk-neutral (or risk-seeking), they will only choose the target (given the payoffs and the fact that level-0 row players choose randomly with equal probabilities). This is where we introduce our adjustment to the GCH model: We assume that players of level-k ($k \ge 1$) hold heterogeneous risk preferences.¹⁹ More specifically, we require that, at least some of these players, exhibit risk aversion. A risk-averse level-1 column player may choose the other strategy (not the target) in the base game. This means that in the extensions, moderately risk-averse level-1 column players may switch to play the target given the increased choice probability of the target by level-0 row players. Note that level-1 row players do not alter their behavior when the base games are extended since level-0 column players’ behavior remains the same as noted above.

Level-2 row players react to level-1 and level-0 column players. The latter do not change their behavior across base games and extensions while the former do - they tend to choose the target more often in the extensions. Thus, level-2 row players choose the target strategy in the extensions with a higher probability than in the base games (the extent to which the target strategy’s choice probability increases depends on their own risk preferences as well as their belief regarding the proportion of the lower hierarchies that they are playing against). Finally, Level-2 row players may also choose the target more frequently in the extensions as long as they believe that they are playing a non-negligible proportion of level-0 row players (since level-1 row players do not alter their behavior). Notice that since the target strategies support an equilibrium, higher levels choose the strategies that constitute that equilibrium with a higher probability in the extension than in the base game.

1.1.1.2 Single-equilibrium games

Level-0 row players tend to choose their target more often in the extensions compared to the base games just like in coordination games. Level-0 column players have a dominating strategy in the base games and in the extensions (which belongs to the never-worst set) and hence they make the same choices across base games and extensions.²⁰

Level-1 row players react to level-0 column players and therefore do not change their behavior across base games and extensions. Level-1 column players have a dominant strategy and therefore their behavior also doesn’t change in the extensions compared to the base games. Finally, level-2 row players will also choose similarly since the behavior of the lower-level column players remains the same, while level-2 column players will once again stick with their dominating strategy. The same arguments apply for higher levels.

1.1.1.3 Taking stock–predictions of GCH

The GCH model predicts more choices of the target strategy by row players in the extensions in both coordination games and single-equilibrium games. In the former, this is due to level-0 players’ reaction to the extension as well as level-2 row players while in the latter this is only due to level-0 players’ reaction. As for the column players, with some degree of risk aversion of players, the model predicts more choices of the target in the extended coordination games but no difference in their behavior in single-equilibrium base and extended games.

Thus, the model predicts the findings well with one caveat—we do not find more choices of the target by row players in single-equilibrium games. In order to reconcile this gap within the framework of the GCH model, one possibility is to consider that there is a very small amount of level-0 players in our pool of participants. This is consistent with some studies of the level-k models that found that level-0 exists only in the minds of higher types (Costa-Gomes & Crawford, 2006; Crawford & Iriberri, 2007). Taking this consideration into account, we get a very minor effect of the extensions on row players in the single-equilibrium games but an effect remains for coordination games (due to the effect on players of level-2). Put differently, accepting that our sample comprises only a negligible proportion of level-0 players and that the remaining participants exhibit some level of risk aversion, GCH provides a comprehensive explanation for our finding

1.1.2 Relevant Strategies and the GCH Model

1.1.2.1 Coordination games

According to the GCH model, the compromise strategy (Middle) belongs to the never-worst set and therefore its choice share relative to Bottom increases (compared to the base game) by level-0 row players’ behavior.²¹Level-0 column players choose randomly (50-50) as in the base game since no strategy belongs to the never worst set. Consequently, level-1 row players will behave as in the base game and will choose the strategy that fits their level of risk aversion. Level-1 column players with some degree of risk aversion will react to the shift in behavior of level-0 row players and the model predicts a higher share of target choices in the extension with the extreme strategy (the same applies to level-2 column players who react to both level-1 and level-0 row players). Finally, level-2 row players’ behavior may be affected in the direction of more choices of the extreme strategy by their reaction to level-1 column players.

Taking the two considerations that we took earlier—some degree of risk aversion and a negligible amount of level-0 players—we obtain a reaction from the column players leading to more target choices in the presence of the extreme strategy but a weak to negligible reaction to its presence by row players. These predictions fit quite well with our finding as there is no direct effect of the added strategy on row players but some positive effect on the column players, i.e., an indirect effect.

1.1.2.2 Single-equilibrium games

Level-0 row players react to this addition similarly to their reaction in coordination games. Level-0 column players do not react since they simply choose their dominant strategy which belongs to the never-worst set.²²Level-1 row players act similarly to the base game since nothing changes in the behavior of level-0 column players. Level-1 column players don’t alter their behavior since they have a dominant strategy (which also holds true for level-2 column players). Finally, level-2 row players do not change their behavior due to the unchanged behavior of the lower-level column players.

Overall, the model predicts no difference in the behavior of either player due to the addition of the extreme strategy in the single-equilibrium games. This holds true in the data for the row players but is at odds with the observed behavior of the column players. They choose their dominant strategy more frequently in the presence of the extreme strategy compared to the base game. While this may seem puzzling, keep in mind that in the base game, there is a non-trivial trade-off between following the dominant strategy and choosing the action that may lead to the surplus maximizing payoff. The presence of the extreme strategy makes choosing the dominated strategy harder to justify for the column player since it may lead to the surplus minimizing payoff. This may be the force that pushes the column players away from this strategy. Note that the above considerations are outside the scope of the GCH model (or any other model that ignores other-regarding preferences).

1.2 Appendix C2: Additional theoretical models

We discuss three more models that allow seemingly irrelevant strategies to affect behavior. Following a brief outline of each model’s main components, we examine to what extent it is able to accommodate the choice patterns that show up in our experiment. That is, we check whether it predicts an increase in target choices by both players when an irrelevant strategy is added to coordination games but no effect of such an addition in single-equilibrium games. While the first presented model, an adjusted level-k model, is successful in explaining our findings, the latter two approaches, QRE and sampling equilibrium, are not.

An adjusted level- k model

A standard level-k model, where each level responds only to level-$k-1$ may also explain our findings as long as the level-0 types are attracted to the salient features that appear in our games’ extensions. Taking this approach is in line with a substantial strand of the level-k literature that assumes a level-0 type who is attracted to salient strategies (e.g., Crawford & Iriberri, 2007; Arad, 2012; Arad & Rubinstein, 2012; Hargreaves Heap et al., 2014; Alaoui & Penta, 2016). A simple exercise, that we exclude for brevity, shows that under the same assumptions (risk aversion and a negligible amount of level-0 players) this model derives predictions that are similar to those derived above for the GCH model.²³

Quantal response equilibrium (QRE)

(McKelvey & Palfrey, 1995). This concept is a generalization of Nash Equilibrium that allows for errors in players’ optimizations. Given an error structure, a player’s probability of choosing a given action is equal to the probability that the action is optimal given his belief regarding his opponents’ strategies. In a QRE, the players’ beliefs are correct.

In most of the theory’s applications, players’ errors are assumed to be i.i.d across strategies, and every error is drawn from an extreme value distribution. This specification leads to the logistic quantal response function in which the probability of player i choosing strategy j is given by

$$\begin{aligned} p_{ij}=\frac{e^{\lambda \bar{u}_{ij}(p_{-i})}}{ \sum \limits _{k} e^{\lambda \bar{u}_{ik}(p_{-i})}} \end{aligned}$$

where $\bar{u}_{ij} (p_{-i})$ is the utility for player i when he chooses action j given that other players are playing according to the probability distribution $p_{-i}.$

The QRE model with the above response function accommodates some of our findings. For example, it predicts that in coordination games a strategy will be chosen more often when it is duplicated compared to when it is not. However, consider a duplicated strategy in a single-equilibrium game. The data shows that the column player in the duplicates extension maintains similar choice probabilities between Left and Right as in the base game. If the model is required to fit the column players’ observed behavior, then it must predict that the row players choose the target strategy more often in the duplicates extension (i.e., Up or Middle) than in the base game (Up), in contrast to our findings. Thus, the model cannot account for our findings as a whole in the single-equilibrium games. In addition, the logistic response function assigns a non-negligible probability to choosing the added dominated strategy in the dominance extensions (especially when the dominated strategy yields payoffs which are only slightly lower than those of the dominating strategy as in our experiment). This feature of the model is at odds with our findings, as row players almost never chose the added dominated strategies.

Sampling equilibrium

(Osborne & Rubinstein, 1998). According to this concept, a player behaves as if he sampled each of his actions once, observed the outcome of playing the sampled action against a random player from the population, and chose the strategy which was associated with the highest payoff. In a sampling equilibrium, the probability with which a player chooses an action is the probability with which that action achieves the highest payoff.

This procedure is unable to generate precise predictions in our setup as it allows for multiple equilbria in our base coordination games and in their extensions. For example, in the duplicates extension, the row player choosing Up with probability p and the column player playing Right with probability p is a sampling equilibrium of the coordination base games, for any $p \in [0,1]$. In their duplicates extension, we get a similar set of equilibria: Row players choose Up and Middle, each with probability p/2, and the column players choose right with probability p. Thus, this multiplicity of equilibria may explain the pattern of our comparative statics, but it may also explain any other pattern. At a broader level, this equilibrium concept is better suited for situations involving repetition and learning, where individuals can explore their own strategies to understand the optimal course of action. For instance, it is akin to searching for the fastest route to the workplace by experimenting with different routes every day. However, in our experiment, participants do not receive feedback, making the model less appropriate for this specific context.