# Behavior in Strategic Settings: Evidence from a Million Rock-Paper-Scissors Games

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^{2}

^{3}

^{4}

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^{†}

## Abstract

**:**

^{,}3 Yet it is not obvious how one should use the information provided. Players can use the information to determine whether an opponent’s past play is consistent with Nash, but without seeing what information an opponent was reacting to (they do not observe the past histories of the opponent’s previous opponents), it is hard to guess what non-Nash strategy the opponent may be using. Additionally, players are not shown information about their own past play, so if a player wants to exploit an opponent’s expected reaction, he must keep track of his own history of play.

## 1. Data: Roshambull

## 2. Model

**Proposition**

**1.**

**Proof.**

## 3. Players Respond to Information

## 4. Level-k Behavior

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 4.1. Reduced-Form Evidence for Level-k Play

#### Multinomial Logit

#### 4.2. Maximum Likelihood Estimation of a Structural Model of Level-k Thinking

**Assumption**

**1.**

**Assumption**

**2.**

#### 4.3. Cognitive Hierarchy

**Definition**

**4.**

- randomizes according to the opponent’s historical distribution 79.92% of the time
- chooses (randomly between) the throw(s) that maximize expected payoff against the player’s own historical distribution 20.08% of the time

#### 4.4. Naive Level-k Strategies

**Definition**

**5.**

**Definition**

**6.**

#### 4.5. Comparisons

#### 4.6. When Are Players’ Throws Consistent with ${k}_{1}$?

## 5. (Non-Equilibrium) Quantal Response

## 6. Likelihood Comparison

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Behavior in Strategic Settings: Evidence from a Million Rock-Paper-Scissors Games

#### Appendix A.1. Proof of Proposition 1

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

#### Appendix A.2. Additional Tables

Opp’s Historical % | Throws (%) | N | ||
---|---|---|---|---|

Paper | Rock | Scissors | ||

0%–25% | 38.88 | 38.26 | 22.87 | 904,003 |

25%–30% | 36.93 | 39.04 | 24.03 | 568,683 |

30%–$33\frac{1}{3}$% | 35.95 | 37.56 | 26.49 | 475,438 |

$33\frac{1}{3}$%–37% | 35.23 | 34.39 | 30.37 | 888,846 |

37%–42% | 32.86 | 29.76 | 37.39 | 671,106 |

42%–100% | 29.91 | 28.03 | 42.06 | 1,173,439 |

Opp’s Historical % | Throws (%) | N | ||
---|---|---|---|---|

Paper | Rock | Scissors | ||

0%–25% | 39.61 | 28.05 | 32.34 | 1,308,484 |

25%–30% | 37.96 | 28.07 | 33.98 | 730,391 |

30%–$33\frac{1}{3}$% | 35.92 | 29.67 | 34.41 | 546,698 |

$33\frac{1}{3}$%–37% | 32.85 | 34.29 | 32.86 | 825,972 |

37%–42% | 28.77 | 40.99 | 30.24 | 513,185 |

42%–100% | 27.24 | 46.64 | 26.12 | 756,785 |

Covariate | Dependent Var: Dummy for Throwing Rock | ||
---|---|---|---|

(1) | (2) | (3) | |

Opp’s Fraction Paper (first) | −0.0403 *** | −0.0718 *** | −0.0952 *** |

(0.0021) | (0.0056) | (0.0087) | |

Opp’s Fraction Scissors (first) | 0.2375 *** | 0.1552 *** | 0.1377 *** |

(0.0022) | (0.0060) | (0.0094) | |

Opp’s Fraction Paper (all) | 0.0022 | 0.0249 ** | 0.0032 |

(0.0032) | (0.0088) | (0.0138) | |

Opp’s Fraction Scissors (all) | 0.0403 *** | 0.0238 * | −0.0187 |

(0.0033) | (0.0093) | (0.0146) | |

Opp’s Paper Lag | 0.0043 *** | −0.0016 | −0.0043 * |

(0.0007) | (0.0015) | (0.0019) | |

Opp’s Scissors Lag | 0.0142 *** | 0.0052 ** | −0.0018 |

(0.0008) | (0.0016) | (0.0020) | |

Own Fraction Paper (first) | −0.0657 *** | 0.0290 *** | −0.1092 *** |

(0.0020) | (0.0073) | (0.0176) | |

Own Fraction Scissors (first) | −0.0816 *** | 0.0013 | −0.3216 *** |

(0.0021) | (0.0076) | (0.0180) | |

Own Fraction Paper (all) | −0.0184 *** | −0.0551 *** | −0.1408 *** |

(0.0030) | (0.0113) | (0.0244) | |

Own Fraction Scissors (all) | −0.0097 ** | −0.0781 *** | −0.0691 ** |

(0.0032) | (0.0119) | (0.0253) | |

Own Paper Lag | −0.0015 * | 0.0121 *** | 0.0046 * |

(0.0007) | (0.0015) | (0.0019) | |

Own Scissors Lag | 0.0031 *** | 0.0094 *** | −0.0059 ** |

(0.0008) | (0.0016) | (0.0019) | |

Constant | 0.3423 *** | −0.0190 *** | 0.0001 |

(0.0007) | (0.0014) | (0.0018) | |

${R}^{2}$ | 0.0184 | ||

N | 4,433,260 |

Covariate | Dependent Var: Dummy for Throwing Paper | ||
---|---|---|---|

(1) | (2) | (3) | |

Opp’s Fraction Rock (first) | 0.2554 *** | 0.1441 *** | 0.1249 *** |

(0.0021) | (0.0058) | (0.0091) | |

Opp’s Fraction Scissors (first) | −0.0363 *** | −0.0648 *** | −0.1095 *** |

(0.0022) | (0.0060) | (0.0095) | |

Opp’s Fraction Rock (all) | 0.0682 *** | 0.0109 | 0.0054 |

(0.0032) | (0.0091) | (0.0144) | |

Opp’s Fraction Scissors (all) | −0.0079 * | −0.0090 | 0.0070 |

(0.0034) | (0.0096) | (0.0153) | |

Opp’s Rock Lag | 0.0112 *** | 0.0048 ** | −0.0047 * |

(0.0007) | (0.0016) | (0.0020) | |

Opp’s Scissors Lag | 0.0037 *** | −0.0029 | −0.0051 ** |

(0.0007) | (0.0016) | (0.0019) | |

Own Rock Lag | −0.0070 *** | 0.0102 *** | −0.0046 * |

(0.0007) | (0.0015) | (0.0019) | |

Own Scissors Lag | 0.0023 *** | 0.0127 *** | 0.0012 |

(0.0007) | (0.0015) | (0.0019) | |

Constant | 0.3365 *** | −0.0097 *** | 0.0065 *** |

(0.0006) | (0.0014) | (0.0018) | |

${R}^{2}$ | 0.0216 | ||

N | 4,433,260 |

Covariate | Dependent Var: Dummy for Throwing Scissors | ||
---|---|---|---|

(1) | (2) | (3) | |

Opp’s Fraction Paper (first) | 0.2395 *** | 0.1637 *** | 0.1241 *** |

(0.0022) | (0.0061) | (0.0095) | |

Opp’s Fraction Rock (first) | −0.0541 *** | −0.0533 *** | −0.0963 *** |

(0.0021) | (0.0057) | (0.0090) | |

Opp’s Fraction Paper (all) | 0.0326 *** | −0.0118 | −0.0171 |

(0.0033) | (0.0095) | (0.0151) | |

Opp’s Fraction Rock (all) | −0.0345 *** | 0.0031 | −0.0192 |

(0.0032) | (0.0090) | (0.0143) | |

Opp’s Paper Lag | 0.0123 *** | 0.0045 ** | −0.0025 |

(0.0007) | (0.0016) | (0.0020) | |

Opp’s Rock Lag | 0.0063 *** | −0.0022 | −0.0021 |

(0.0007) | (0.0016) | (0.0019) | |

Own Paper Lag | 0.0049 *** | 0.0096 *** | −0.0085 *** |

(0.0007) | (0.0015) | (0.0019) | |

Own Rock Lag | −0.0052 *** | 0.0233 *** | 0.0011 |

(0.0007) | (0.0015) | (0.0019) | |

Constant | 0.3034 *** | 0.0000 | 0.0047 ** |

(0.0006) | (0.0014) | (0.0018) | |

${R}^{2}$ | 0.0214 | ||

N | 4,433,260 |

Opponent’s Expected Payoff of Paper | Opponent’s Throw (%) | N | ||
---|---|---|---|---|

Paper | Rock | Scissors | ||

$[-1,\phantom{\rule{0.166667em}{0ex}}-0.666]$ | 33.74 | 34.52 | 31.74 | 2448 |

$[-0.666,\phantom{\rule{0.166667em}{0ex}}-0.333]$ | 36.5 | 34.06 | 29.44 | 11,126 |

$[-0.333,\phantom{\rule{0.166667em}{0ex}}0]$ | 34.77 | 33.78 | 31.45 | 836,766 |

$[0,\phantom{\rule{0.166667em}{0ex}}0.333]$ | 34.27 | 33.38 | 32.35 | 1,358,196 |

$[0.333,\phantom{\rule{0.166667em}{0ex}}0.666]$ | 32.57 | 34.57 | 32.86 | 9867 |

Opponent’s Expected Payoff of Scissors | Opponent’s Throw (%) | N | ||
---|---|---|---|---|

Paper | Rock | Scissors | ||

$[-1,\phantom{\rule{0.166667em}{0ex}}-0.666]$ | 34.24 | 32.49 | 33.27 | 1659 |

$[-0.666,\phantom{\rule{0.166667em}{0ex}}-0.333]$ | 33.72 | 34.97 | 31.31 | 19,249 |

$[-0.333,\phantom{\rule{0.166667em}{0ex}}0]$ | 35.46 | 32.02 | 32.52 | 1,041,591 |

$[0,\phantom{\rule{0.166667em}{0ex}}0.333]$ | 33.6 | 34.81 | 31.59 | 1,139,164 |

$[0.333,\phantom{\rule{0.166667em}{0ex}}0.666]$ | 31.73 | 40.02 | 28.24 | 16,740 |

(1) | (2) | (3) | |
---|---|---|---|

Payoff from Playing K2 | −0.021 *** | 0.159 *** | 0.145 *** |

(0.0010) | (0.0015) | (0.0019) | |

High Opponent Experience | 0.005 *** | ||

(0.0014) | |||

Medium Opponent Experience | −0.001 | ||

(0.0012) | |||

K2 Payoff X High Opponent Experience | 0.052 *** | ||

(0.0037) | |||

K2 Payoff X Medium Opponent Experience | 0.045 *** | ||

(0.0029) | |||

Experienced | 0.006 *** | ||

(0.0011) | |||

Exp X Payoff from Playing K2 | −0.012 ** | −0.085 *** | −0.093 *** |

(0.0047) | (0.0047) | (0.0065) | |

Exp X High Opponent Experience | 0.006 ** | ||

(0.0027) | |||

Exp X Medium Opponent Experience | 0.008 *** | ||

(0.0027) | |||

Exp X K2 Payoff X High Opponent Experience | −0.027 ** | ||

(0.011) | |||

Exp X K2 Payoff X Medium Opponent Experience | −0.036 *** | ||

(0.010) | |||

Own Games>100 | 0.002 | ||

(0.0018) | |||

Own Games>100 X Payoff from Playing K2 | −0.097 *** | 0.144 *** | 0.119 *** |

(0.021) | (0.013) | (0.022) | |

Own Games>100 X High Opponent Experience | −0.024 *** | ||

(0.0025) | |||

Own Games>100 X Medium Opponent Experience | −0.025 *** | ||

(0.0031) | |||

Own Games>100 X K2 Payoff X High Opponent Experience | 0.170 *** | ||

(0.028) | |||

Own Games>100 X K2 Payoff X Medium Opponent Experience | 0.180 *** | ||

(0.033) | |||

Player Fixed Effects | No | Yes | Yes |

Observations | 4,130,024 | 4,130,024 | 4,130,024 |

Adjusted ${R}^{2}$ | 0.000 | 0.003 | 0.004 |

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1 | Two players each play rock, paper, or scissors. Rock beats scissors; scissors beats paper; paper beats rock. If they both play the same, it is a tie. The payoff matrix is in Section 2. |

2 | If the opponent is not playing Nash, then Nash is no longer a best response. In symmetric zero-sum games such as RPS, deviating from Nash is costless if the opponent is playing Nash (since all strategies have an expected payoff of zero), but if a player thinks he knows what non-Nash strategy his opponent is using then there is a profitable deviation from Nash. |

3 | Work in evolutionary game theory on RPS has looked at how the population’s distribution of strategies evolves towards or around Nash equilibrium (e.g., [7,8]). Past work on fictitious play has showed that responding to the opponents’ historical frequency of strategies leads to convergence to Nash equilibrium [9,10,11]. Young [12] also studies how conventions evolve as players respond to information about how their opponents have behaved in the past, while Mookherjee and Sopher [13,14] examine the effect of information on opponents’ history on strategic choices. |

4 | When doing a test at the player level, we expect about 5% of players to be false positives, so we take these numbers as evidence on behavior only when they are statistically significant for substantially more than 5% of players. |

5 | Since the focal ${k}_{0}$ strategies can be skewed, our ${k}_{1}$ and ${k}_{2}$ strategies usually designate a unique throw, which would not be true if ${k}_{0}$ were constrained to be a uniform distribution. |

6 | As we discuss in Section 4, there are several reasons that may explain why we find lower estimates for ${k}_{1}$ and ${k}_{2}$ play than in previous work. Many players may not remember their own history, which is necessary for playing ${k}_{2}$. Also, given that ${k}_{0}$ is what players would most likely play if they were not shown the information (i.e., when they play RPS outside the application), it may be more salient than in other contexts. |

7 | Because we think players differ in the extent to which they respond to information and consider expected payoffs, we do not impose the restriction from quantal response equilibrium theory [30] that the perceived expected payoffs are correct. Instead, we require that the expected payoffs are calculated based on the history of play. See Section 5 for more detail. |

8 | RPS is usually played for low stakes, but sometimes the result carries with it more serious ramifications. During the World Series of Poker, an annual $500 per person RPS tournament is held, with the winner taking home $25,000. RPS was also once used to determine which auction house would have the right to sell a $12 million Cezanne painting. Christie’s went to the 11-year-old twin daughters of an employee, who suggested “scissors” because “Everybody expects you to choose ‘rock’.” Sotheby’s said that they treated it as a game of chance and had no particular strategy for the game, but went with “paper” [31]. |

9 | Bart Johnston, one of the developers said, “We’ve added this intriguing statistical aspect to the game…You’re constantly trying to out-strategize your opponent” [32]. |

10 | Unfortunately we only have a player id for each player; there is no demographic information or information about their out-of-game connections to other players. |

11 | Because of the possibility for players to collude to give one player a good record if the other does not mind having a bad one, we exclude matches from the small fraction of player-pairs for which one player won an implausibly high share of the matches (100% of ≥10 games or 80% of ≥20 games). To accurately recreate the information that opponents were shown when those players played against others, we still include those “collusion” matches when forming the players’ histories. |

12 | Depending on the opponent’s history, the strategies we look at may not indicate a unique throw (e.g., if rock and paper have the same expected payoffs); for some analyses we only use players who have 100 clean matches where the strategies being considered indicate a unique throw, so we use between 5405 and 7758 players. |

13 | Players could use aspects of their history that are not observable to the opponent as a private randomization devices, but conditional on all information available to the opponent, they must be mixing $\frac{1}{3},\frac{1}{3},\frac{1}{3}$. |

14 | We also find serial correlation both across throws within a match and across matches, which is inconsistent with Nash equilibrium. |

15 | Inexperienced players also have a lot of variance in the fraction of time they play rock, but for them it is hard to differentiate between deviations from $\frac{1}{3},\frac{1}{3},\frac{1}{3}$ and noise from randomization. |

16 | If all players were playing Nash, we would expect to reject the null for 5% of players; with 95% probability we would reject the null for less than 5.44% of players. |

17 | If players were truly, strictly maximizing their payoff against the opponent’s past distribution, this change would be even more stark, though it would not go exactly from 0 to 1 since the optimal response also depends on the percent of paper (or scissors) played, which the table does not condition on. |

18 | The Appendix A.2 has the same table adding own history. The coefficients on opponent’s history are basically unaffected. The coefficients on own history reflect the imperfect randomization—players who played rock in the past are more likely to play rock. |

19 | If we run the regression with just the distribution of all throws or just the lags, the signs are as expected, but that seems to be mostly picking up the effect via the opponent’s distribution of first throws. |

20 | The reduced-form results indicate that players react much more strongly to the distribution of first throws than to the other information provided. |

21 | Alternatively, a ${k}_{1}$ player may think that ${k}_{0}$ is strategic, but playing an unknown strategy so past play is the best predictor of future play. |

22 | Nash always has an expected payoff of zero. As show in Table 5, best responding can have an expected payoff of 14¢ for every dollar bet. |

23 | Sometimes opponents’ distributions are such that there are multiple throws that are tied for the highest expected payoff. For our baseline specification we ignore these throws. As a robustness check we define alternative ${k}_{1}$-strategies where one throw is randomly chosen to be the ${k}_{1}$ throw when payoffs are tied or where both throws are considered consistent with ${k}_{1}$ when payoffs are tied. The results do not change substantially. |

24 | In the proof of Proposition 1 we show that in Nash equilibrium, histories do not affect continuation values, so in equilibrium it is a result, not an assumption, that players are myopic. However, out of Nash equilibrium, it is possible that what players throw now can affect their probability of winning later rounds. |

25 | One statistic that we thought might affect continuation values is the skew of a player’s historical distribution. As a player’s history departs further from random play, there is more opportunity for opponent response and player exploitation of opponent response. We ran multinomial logits for each experienced player on the effect of own history skewness on the probability of winning, losing, or drawing. The coefficients were significant for less than (the expected false positives of) 5% of players. This provides some support to our assumption that continuation values are not a primary concern. |

26 | As an aside, in the case of RPS the level ${k}_{j+6}$ strategy is identical to the level ${k}_{j}$ strategy for $j\ge 1$, so it is impossible to identify levels higher than 6. One might expect ${k}_{j}$ to be equivalent to ${k}_{j+3}$, but ${k}_{1}$, ${k}_{3}$ and ${k}_{5}$ strategies depend on the opponent’s history, with one being rock, one being paper, and one being scissors, while levels ${k}_{2}$, ${k}_{4}$ and ${k}_{6}$ strategies depend on one’s own history. So, with many games all strategies ${k}_{j}$ with $j<7$ are separately identified. This also implies that the ${k}_{1}$ play we observe could in fact be ${k}_{7}$ play, but we view this as highly unlikely. |

27 | Since we do the analysis within player, the estimates would be very imprecise for players with fewer games. |

28 | We derive the Hessian of the likelihood function, plug in the estimates, and take the inverse. |

29 | Other work, such as [35], has found evidence of players mixing levels of sophistication across different games. |

30 | To fully calculate the equilibrium, we could repeat the analysis using the frequencies of $c{h}_{0}$ and $c{h}_{1}$ found below and continue until the frequencies converged, but since the estimated $c{h}_{0}$ and $c{h}_{1}$ are near the 79% and 20% we started with, we do not think this computationally intense exercise would substantially change the results. |

31 | Wright and Leyton-Brown [40]’s estimates are closer to ours, with 95% confidence interval for the average number of iterative steps of [0.51, 0.59]. |

32 | Similar predictions could be made about ${k}_{2}$ play; however, since we find that ${k}_{2}$ is used so little, we do not model ${k}_{2}$ play in this section. See Appendix A.2 |

33 | We present OLS coefficients instead of logit, so they will be comparable when we add fixed-effects, which do not work with logit. When running logits at the player level, the median coefficients are in the range of the above coefficients, but the model is totally lacking power—the coefficients are significant for less than 5% of players. |

34 | If we allowed the expected payoff to vary or be estimated from the data, this would increase the flexibility of the model, making any improvement in fit suspect. However, since the expected payoff is completely determined by the opponent’s historical distribution the model has only 3 free parameters—estimated separately for each player—the $\beta $ and two $\alpha $s. (The third $\alpha $ is normalized to zero.) |

35 | A second level of reasoning would expect opponents to play according to the distribution induced by one’s own history and would play with probabilities proportional to the expected payoff against that distribution. However, given the low levels of ${k}_{2}$ play we find and the econometric difficulties of including own history in the logit, we only analyze the first iteration of reasoning. |

36 | This suggests that players do respond to expected payoffs calculated from historical opponent play; whereas in the reduced-form results (Table 4) we showed that players did not respond to the expected payoff calculated from predicted opponent play—predicted based on the coefficients from Table 3 and players’ histories. |

37 | We multiply by 100 to convert to percentages and by 2/9 to evaluate the margin at the mean: $0.232\xb71.42\xb7100\xb7\frac{2}{9}=7.3$. |

38 | Models with additional parameters have more flexibility to fit data better even when the underlying model is no better. We find little evidence of ${k}_{2}$ and using a ${k}_{1}$ model avoids choosing how much to penalize the level-k model for the additional parameter. |

39 | This would be an interesting dimension for future work to explore. |

40 | In the actual game players could challenge a specific player to a game or be matched by the software to someone else who was looking for an opponent. |

41 | Because what matters for the result is the symmetry across strategies at all stages, having an intra-match discount factor does not change the result, but substantially complicates the proof. |

**Figure 2.**Number of matches played by Roshambull users. Note: The figure shows the number of clean matches played by the 334,661 players who had at least one clean, completed match (see Footnote 11 for a description of the data cleaning). The data has a very long right tail, so all the players with over 200 matches are grouped together in the right-most bar.

**Figure 4.**Percent of last 100 Throws that are Rock—Observed and Predicted. Note: For each of the 7758 players with at least 100 matches we calculate the percent of his or her last 100 throws that were rock (purple distribution). We overlay the binomial distribution with $n=100$ and $p=\frac{1}{3}$.

**Figure 5.**Level-k consistency. Note: These graphs show the distribution across the 6674 players who have 100 games with uniquely defined ${k}_{1}$ and ${k}_{2}$ strategies of the fraction of throws that are ${k}_{1}$- and ${k}_{2}$-consistent. The vertical line indicates $\frac{1}{3}$, which we would expect to be the mean of the distribution if throws were random. (

**a**) Percent of player’s throws that are ${k}_{1}$-consistent. (

**b**) Percent of player’s throws that are ${k}_{2}$-consistent.

**Figure 6.**Coefficient in the ${k}_{1}$ multinomial logit. Note: The coefficient is $\beta $ from the logit estimation, run separately for each player, ${U}_{i}^{rock}={\alpha}^{rock}+\beta \xb7{k}_{1,i}^{rock}+{\u03f5}_{i}^{rock}$, analogously for paper and scissors, where ${k}_{1,}^{rock}$ is a dummy for whether rock is the ${k}_{1}$-consistent thing to do on throw i. Outliers more than 4 standard deviations from the mean are omitted.

**Figure 7.**Distribution across players of the coefficient in the quantal response model. Note: The coefficient is $\beta $ from the logit estimation, run separately for each player, ${U}_{i}^{rock}={\alpha}^{rock}+\beta \xb7{\mathrm{EP}}^{rock}+{\u03f5}_{i}^{rock}$, analogously for paper and scissors, where EP (the expected payoff) for the baseline model is the probability of winning against the opponent’s historical distribution minus the probability of losing; for the naive model is just the probability of winning. Outliers more than 4 standard deviations from the mean are omitted.

**Figure 8.**Probability of the level-${k}_{1}$ model. Note: For each player we calculate the probability that the data were generated by the quantal response model as opposed to the level-${k}_{1}$ model, assuming a flat prior, as in Equation (2).

Variable | Full Sample | Restricted Sample | ||
---|---|---|---|---|

Mean | (SD) | Mean | (SD) | |

Throw Rock (%) | 33.99 | (47.37) | 32.60 | (46.87) |

Throw Paper (%) | 34.82 | (47.64) | 34.56 | (47.56) |

Throw Scissors (%) | 31.20 | (46.33) | 32.84 | (46.96) |

Player’s Historical %Rock | 34.27 | (19.13) | 32.89 | (8.89) |

Player’s Historical %Paper | 35.14 | (18.80) | 34.88 | (8.97) |

Player’s Historical %Scissors | 30.59 | (17.64) | 32.23 | (8.57) |

Opp’s Historical %Rock | 34.27 | (19.13) | 33.59 | (13.71) |

Opp’s Historical %Paper | 35.14 | (18.80) | 34.76 | (13.46) |

Opp’s Historical %Scissors | 30.59 | (17.64) | 31.65 | (12.84) |

Opp’s Historical Skew | 10.42 | (18.08) | 5.39 | (12.43) |

Opp’s Historical %Rock (all throws) | 35.45 | (12.01) | 34.81 | (8.59) |

Opp’s Historical %Paper (all throws) | 34.01 | (11.74) | 34.10 | (8.35) |

Opp’s Historical %Scissors (all throws) | 30.54 | (11.07) | 31.09 | (7.98) |

Opp’s Historical Length (matches) | 55.92 | (122.05) | 99.13 | (162.16) |

Total observations | 5,012,128 | 1,472,319 |

Opp’s Historical % | Throws (%) | N | ||
---|---|---|---|---|

Paper | Rock | Scissors | ||

0%–25% | 27.07 | 35.84 | 37.09 | 1,006,726 |

25%–30% | 27.11 | 35.23 | 37.66 | 728,408 |

30%–$33\frac{1}{3}$% | 29.87 | 34.72 | 35.41 | 565,623 |

$33\frac{1}{3}$%–37% | 34.21 | 34.44 | 31.35 | 794,886 |

37%–42% | 40.45 | 32.58 | 26.97 | 529,710 |

42%–100% | 46.58 | 30.35 | 23.06 | 1,056,162 |

Covariate | Dependent Var: Dummy for Throwing Rock | ||
---|---|---|---|

(1) | (2) | (3) | |

Opp’s Fraction Paper (first) | −0.0382 *** | −0.0729 *** | −0.0955 *** |

(0.0021) | (0.0056) | (0.0087) | |

Opp’s Fraction Scissors (first) | 0.2376 *** | 0.1556 *** | 0.1381 *** |

(0.0022) | (0.0060) | (0.0094) | |

Opp’s Fraction Paper (all) | 0.0011 | 0.0258 ** | 0.0033 |

(0.0032) | (0.0088) | (0.0138) | |

Opp’s Fraction Scissors (all) | 0.0416 *** | 0.0231 * | −0.0208 |

(0.0033) | (0.0093) | (0.0146) | |

Opp’s Paper Lag | 0.0052 *** | −0.0019 | −0.0043 * |

(0.0007) | (0.0015) | (0.0019) | |

Opp’s Scissors Lag | 0.0139 *** | 0.0055 *** | −0.0017 |

(0.0008) | (0.0016) | (0.0020) | |

Own Paper Lag | −0.0171 *** | 0.0239 *** | 0.0051 ** |

(0.0007) | (0.0015) | (0.0019) | |

Own Scissors Lag | −0.0145 *** | 0.0208 *** | −0.0047 * |

(0.0007) | (0.0015) | (0.0019) | |

Constant | 0.3548 *** | −0.0264 *** | −0.0009 |

(0.0006) | (0.0014) | (0.0018) | |

${R}^{2}$ | 0.0172 | ||

N | 4,433,260 |

Opponent’s Expected Payoff of Rock | Opponent’s Throw (%) | N | ||
---|---|---|---|---|

Paper | Rock | Scissors | ||

$[-1,\phantom{\rule{0.166667em}{0ex}}-0.666]$ | 30.83 | 40.56 | 28.61 | 2754 |

$[-0.666,\phantom{\rule{0.166667em}{0ex}}-0.333]$ | 32.89 | 38.56 | 28.55 | 65,874 |

$[-0.333,\phantom{\rule{0.166667em}{0ex}}0]$ | 33.83 | 34 | 32.17 | 1,266,538 |

$[0,\phantom{\rule{0.166667em}{0ex}}0.333]$ | 35.52 | 32.45 | 32.03 | 871,003 |

$[0.333,\phantom{\rule{0.166667em}{0ex}}0.666]$ | 34.4 | 34.47 | 31.13 | 12,234 |

Wins (%) | Draws (%) | Losses (%) | Wins (%) − Losses (%) | N | |
---|---|---|---|---|---|

Full Sample | 33.8 | 32.4 | 33.8 | 0 | 5,012,128 |

Experienced Sample | 34.66 | 32.17 | 33.17 | 1.49 | 1,472,319 |

Best Response to Predicted Play | 41.66 | 30.97 | 27.37 | 14.29 | 2,218,403 |

Variable | Definition |
---|---|

${\widehat{k}}_{0}^{r}$ | fraction of the time a player plays ${k}_{0}$ and chooses rock |

${\widehat{k}}_{0}^{p}$ | fraction of the time a player plays ${k}_{0}$ and chooses paper |

${\widehat{k}}_{0}^{s}$ | fraction of the time a player plays ${k}_{0}$ and chooses scissors |

${\widehat{k}}_{1}$ | fraction of the time a player plays ${k}_{1}$ |

$({\widehat{k}}_{2})$ | $1-{\widehat{k}}_{1}-{\widehat{k}}_{0}^{r}-{\widehat{k}}_{0}^{p}-{\widehat{k}}_{0}^{s}$ (not an independent parameter) |

Variable | Mean | SD | Median | Min | Max |
---|---|---|---|---|---|

${k}_{0}$ | 0.738 | 0.16 | 0.75 | 0.19 | 1.00 |

${k}_{1}$ | 0.185 | 0.14 | 0.16 | 0.00 | 0.77 |

${k}_{2}$ | 0.077 | 0.08 | 0.06 | 0.00 | 0.41 |

N = 6639 |

Variable | 95% CI Does Not Include 0 | 95% CI Does Not Include 1 | 95% CI Does Not Include 0 or 1 |
---|---|---|---|

${k}_{0}$ | 93.11% | 58.30% | 57.51% |

${k}_{1}$ | 62.87% | 99.97% | 62.87% |

${k}_{2}$ | 11.54% | 95.57% | 11.54% |

N = 6389 |

Variable | Mean | SD | Median | Min | Max |
---|---|---|---|---|---|

${k}_{0}$ | 0.750 | 0.16 | 0.77 | 0.17 | 1.00 |

${k}_{1}$ | 0.161 | 0.14 | 0.14 | 0.00 | 0.77 |

${k}_{2}$ | 0.089 | 0.07 | 0.08 | 0.00 | 0.49 |

N = 6856 |

Variable | Mean | SD | Median | Min | Max |
---|---|---|---|---|---|

${k}_{0}$ | 0.722 | 0.18 | 0.75 | 0.07 | 1.00 |

${k}_{1}$ | 0.211 | 0.17 | 0.18 | 0.00 | 0.93 |

${k}_{2}$ | 0.067 | 0.08 | 0.04 | 0.00 | 0.46 |

N = 5692 |

Model | |||
---|---|---|---|

Level-k | Cognitive Hierarchy | Naive | |

All Players | 798 | 1833 | 2629 |

14.8% | 33.9% | 48.6% | |

Players $>95\%$ Prob | 13 | 188 | 1148 |

1.0% | 13.9% | 85.1% |

(1) | (2) | (3) | |
---|---|---|---|

K1 Payoff | 0.068 *** | 0.105 *** | 0.071 *** |

(0.0012) | (0.0013) | (0.0016) | |

High Opponent Experience | −0.076 *** | ||

(0.0017) | |||

Medium Opponent Experience | −0.055 *** | ||

(0.0015) | |||

K1 Payoff X High Opponent Experience | 0.528 *** | ||

(0.010) | |||

K1 Payoff X Medium Opponent Experience | 0.318 *** | ||

(0.0050) | |||

Experienced | 0.015 *** | ||

(0.0015) | |||

Exp X Payoff from Playing K1 | 0.068 *** | 0.041 *** | −0.021 *** |

(0.0039) | (0.0038) | (0.0046) | |

Exp X High Opponent Experience | −0.019 *** | ||

(0.0034) | |||

Exp X Medium Opponent Experience | −0.023 *** | ||

(0.0035) | |||

Exp X K1 Payoff X High Opponent Experience | 0.091 *** | ||

(0.021) | |||

Exp X K1 Payoff X Medium Opponent Experience | 0.095 *** | ||

(0.013) | |||

Own Games>100 | −0.002 | ||

(0.0022) | |||

Own Games>100 X Payoff from Playing K1 | 0.082 *** | 0.072 *** | 0.021 *** |

(0.0068) | (0.0050) | (0.0049) | |

Own Games>100 X High Opponent Experience | 0.012 *** | ||

(0.0026) | |||

Own Games>100 X Medium Opponent Experience | 0.022 *** | ||

(0.0035) | |||

Own Games>100 X K1 Payoff X High Opponent Experience | −0.025 | ||

(0.023) | |||

Own Games>100 X K1 Payoff X Medium Opponent Experience | 0.014 | ||

(0.016) | |||

Player Fixed Effects | No | Yes | Yes |

Observations | 4,130,024 | 4,130,024 | 4,130,024 |

Adjusted ${R}^{2}$ | 0.003 | 0.004 | 0.008 |

_{1}-consistent. ‘k

_{1}Payoff’ is the expected payoff to playing k

_{1}if the opponent randomizes according to his history (ranges from 0 to 1). ‘High opp exp’ is a dummy for opponents who have 47 or more past games; ‘Medium opp exp’ is a dummy for opponents with 14 to 46 past games. ‘Experienced’ is a dummy for players who eventually play at ≥ 100 games. ‘Own Games > 100’ indicates the player has already played at least 100 games. The ‘X’ indicates the interaction between the dummies and other covariates.

All Players | Players with Significant Difference | |||
---|---|---|---|---|

Regular ${\mathit{k}}_{1}$ | Naive ${\mathit{k}}_{1}$ | Regular ${\mathit{k}}_{1}$ | Naive ${\mathit{k}}_{1}$ | |

Est. MLE fraction ${k}_{1}$ play | 3.451 *** | 2.707 *** | 3.417 *** | 2.619 *** |

(0.0569) | (0.0524) | (0.0661) | (0.0641) | |

Est. QR coefficient on return to ${k}_{1}$ | −0.284 *** | −0.106 *** | −0.248 *** | −0.0870 *** |

(0.0054) | (0.0029) | (0.0051) | (0.0027) | |

Total Number of Matches (100s) | 0.0200 *** | 0.0168 *** | 0.0238 *** | 0.0131 * |

(0.0036) | (0.0042) | (0.0041) | (0.0051) | |

Constant | 0.173 *** | 0.208 *** | 0.0549 ** | 0.135 *** |

(0.0106) | (0.0120) | (0.0192) | (0.0221) | |

${R}^{2}$ | 0.358 | 0.321 | 0.565 | 0.438 |

N | 6673 | 5731 | 2373 | 2216 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Batzilis, D.; Jaffe, S.; Levitt, S.; List, J.A.; Picel, J.
Behavior in Strategic Settings: Evidence from a Million Rock-Paper-Scissors Games. *Games* **2019**, *10*, 18.
https://doi.org/10.3390/g10020018

**AMA Style**

Batzilis D, Jaffe S, Levitt S, List JA, Picel J.
Behavior in Strategic Settings: Evidence from a Million Rock-Paper-Scissors Games. *Games*. 2019; 10(2):18.
https://doi.org/10.3390/g10020018

**Chicago/Turabian Style**

Batzilis, Dimitris, Sonia Jaffe, Steven Levitt, John A. List, and Jeffrey Picel.
2019. "Behavior in Strategic Settings: Evidence from a Million Rock-Paper-Scissors Games" *Games* 10, no. 2: 18.
https://doi.org/10.3390/g10020018