In Fig. 1 we present the results of the 1PG in a scatter plot. Recall that in this case subjects have to pick two numbers, $(x_i,y_i)$ ; the first number is depicted on the horizontal axis, the second on the vertical axis. The diagonal dashed line marks the points where a subject picked the same number for $x_i$ and $y_i$ . The solid circle indicates subjects who fully solved the game (0,0).

As can be seen, only a minority ( $\approx 31\%$ ) of subjects is able to fully solve the 1PG, i.e., pick zero for both numbers. In the remainder of this paper we will use this ability to fully solve the game as our primary measure of understanding of the structure of the guessing game.

Result 1

Only 31% of our subjects fully understand the one-player guessing game.

Another interesting observation in Fig. 1 is that subjects who play numbers closer together also play numbers closer to the origin. This is relevant, as in the 1PG there are two ways in which a subject (who has not fully solved the game) can improve her payoffs: by picking numbers closer to zero, and/or by picking numbers that are closer to each other. A Spearman test confirms the correlation between higher average of both choices and the distance between them (Spearman $\rho =0.83$ , p value $<0.001$ ). As subjects with high payoffs played both numbers that were close to each other, and to zero, one could interpret the payoffs of the 1PG as a measure of (partial) understanding of the structure of the guessing game. Therefore, we will use the payoffs of the 1PG as a secondary measure to complement to our primary measure of understanding, “Solved 1PG”/“Not solved 1PG”.

Fig. 2 Distribution of choices in the 2PG with and without 1PG experience

Figure 3 shows the decisions of subjects in the 2PG on the vertical axis, and their payoffs for the 1PG on the horizontal axis. Subjects who fully solved the 1PG (solid circles) mostly chose zero in the 2PG (24/25, 96%), and picked significantly lower numbers in the 2PG than subjects who did not fully solve the 1PG (Mann–Whitney U Test, p value $<0.001$ ). In line with this, we also observe that subjects who earn higher payoffs in the 1PG play lower numbers in the 2PG (Spearman $\rho =-0.745$ , p value $<0.001$ )

Fig. 3 Distribution of choices in the 2PG (vertical axis) and payoff in the 1PG (horizontal axis)

Result 2

Subjects with a better understanding of the structure of the one-player guessing game play numbers closer to the Nash Equilibrium in the two-player guessing game.

But, is playing numbers near the Nash Equilibrium the best strategy in the 2PG? To answer this question we construct ${\overline{\Pi }}_i^{2PG}$ . This variable represents the payoff that each subject i would have gotten had she played against the average choice of all other subjects j except herself (i.e., $j\ne i$ ). Formally ${\overline{\Pi }}_i^{2PG}$ is defined as:

(6)

Figure 4 illustrates the relationship of ${\overline{\Pi }}_i^{2PG}$ with both, the payoffs of the 1PG ( ${\Pi }_i^{1PG}$ , left panel) and choice in the 2PG ( $z_i$ , right panel). Interestingly, subjects who fully solved the 1PG don’t have the highest ${\overline{\Pi }}_i^{2PG}$ . This is because they play Nash Equilibrium, when payoffs would have been maximized by playing a number close to 9 as can be seen in the right plot. Overall, subjects who fully solved the game did not earn a significantly different payoff compared to subjects who did not fully solve the game (Mann–Whitney U test p value $=0.465$ ).

Analyzing our secondary measure of understanding of the structure of the game ( ${\Pi }_i^{1PG}$ ) reveals a more nuanced pattern: it appears that there is a non-monotonic relationship between understanding of the structure of the game and expected payoffs in the 2PG (see Fig. 15 in "Electronic supplementary material Appendix C" for a close-up of Fig. 4.). Regressing ${\overline{\Pi }}_i^{2PG}$ on ${\Pi }_i^{1PG}$ and ${\left({\Pi }_i^{1PG}\right)}^2$ yields coefficients that are significantly positive and negative respectively. This gives statistical support to the fitted quadratic function in the left panel of Fig. 4, implying that increased understanding leads to increased expected payoffs, but that this relationship reverses for very high levels of understanding.

Fig. 4 Relationship of payoff in the 1PG ( ${\Pi }_i^{1PG}$ ) and ${\overline{\Pi }}_i^{2PG}$ (left panel). The line in the left panel is a fitted quadratic function. The right panel shows the relationship of choice in the 2PG ( $z_i$ ) and ${\overline{\Pi }}_i^{2PG}$ . In both panels the darker dots indicate subjects who fully solved the 1PG

Table 1 Regression of ${\overline{\Pi }}_i^{2PG}$ on the payoff in the 1PG ( ${\Pi }_i^{1PG}$ ) and its square value ${\left({\Pi }_i^{1PG}\right)}^2$

	${\overline{\Pi }}_i^{2PG}$
${\Pi }_i^{1PG}$	$5.519\ast \ast \ast$
	(0.904)
${\left({\Pi }_i^{1PG}\right)}^2$	$-1.565\ast \ast \ast$
	(0.353)
Constant	$-3.389\ast \ast \ast$
	(0.500)
$N$	80
Adj. $R^2$	0.580
Joint test p value	0.000

Standard errors in parentheses

*p < 0.05; **p < 0.01; ***p < 0.001

Result 3

The relationship between understanding of the structure of the one-player guessing game and payoffs in the two-player guessing game follows a non-monotonic pattern.

On the left panel of Fig. 5, we plot the number of tokens subjects have placed correctly in the belief elicitation task against the payoff in the 1PG. Subjects who fully solved the 1PG placed a larger number of tokens correctly (Mann–Whitney U test, p value $=0.001$ ).Footnote ¹⁵ This result is confirmed by the strong correlation that we observe between the 1PG payoff and the number tokens placed correctly (Spearman $\rho =0.583$ , p value < 0.001). On the right panel of Fig. 5 we plot the distribution of tokens (horizontal axis) against the payoff in the 1PG (vertical axis). While subjects who did not fully solve the 1PG spread out their tokens across most of the strategy space, subjects who fully solved the 1PG expect their counterparts to play numbers closer to the Nash Equilibrium (Mann–Whitney U test, p value $<0.001$ ). Again, the correlation between the distance of tokens to NE and payoffs in the 1PG confirms this result (Spearman $\rho =-0.359$ with p value $=0.001$ ) (Table 1).

To test how the accuracy of beliefs relates to the understanding of the structure of the game, we plot the mean of the belief distribution of each subject against their payoff in the 1PG (vertical axis) on the left panel of Fig. 6.Footnote ¹⁶ The vertical dotted line marks the mean choice across all subjects in the 2PG (13.63). The right panel of Fig. 6 plots the absolute distance of individual mean beliefs to mean 2PG play against earnings in the 1PG. Two things are clear from the graph: First, the mean beliefs of some subjects differ quite a bit from mean actual play in the 2PG. Second, subjects who fully solved the 1PG have a lower absolute difference of their mean beliefs and mean choice of all subjects in the 2PG (Mann–Whitney U test p value $=0.030$ ). This result is supported using our secondary measure of understanding (Spearman $\rho =-0.473$ with p value $<0.001$ ).

Fig. 6 In the left panel we present the relationship between the payoff in the 1PG ( ${\Pi }_i^{1PG}$ ) and the mean value of the distributed tokens (horizontal axis). The vertical dotted line marks the mean of all choices in the 2PG (which is 13.63). The right panel illustrates the relationship between the payoff in the 1PG ( ${\Pi }_i^{1PG}$ , vertical axis) and the absolute distance between mean choice of subjects in the 2PG and the mean value of the distributed tokens

Result 4

Subjects with a better understanding of the structure of the one-player guessing game form more accurate beliefs about their counterparts’ choices in the two-player guessing game.

Additionally, we analyze whether choices in the 2PG are best responses to the stated beliefs (i.e., the token distribution). To do so we compute the choice in the 2PG that would maximize the payoff of a subject conditional on her stated beliefs being correct:

(7) $\begin{array}{c}{z}_i^{\ast }\left(B\right)=\underset{{\tilde{z}}_i}{\text{arg max}}\sum _{j=1}^{20}\frac{b_{\mathrm{ij}}}{19}\left(2,-,0.10,\left({\tilde{z}}_i-\frac{2}{3}\frac{{\tilde{z}}_i+{\overline{b}}_j}{2}\right)\right),\end{array}$

where $z_i^{\ast }\left(B_i\right)$ is the choice of subject i that maximizes her payoffs given her beliefs $B_i=(b_{i1},b_{i2},\dots ,b_{i20})$ , $b_{\mathrm{ij}}$ is the number of tokens that subject i put in bin j, and ${\overline{b}}_j$ is the average value of the bin (so for example, for the first bin [0,4], ${\overline{b}}_1=2$ , for the second [5,9], ${\overline{b}}_2=7$ , etc.).Footnote ¹⁷ We then create an individual variable $\Delta z_i^{\ast }=|z_i-z_i^{\ast }\left(B_i\right)|$ which is the absolute difference between actual choice of subject i in the 2PG minus the optimal choice conditional on her stated beliefs. Figure 7 illustrates the relation of $\Delta z_i^{\ast }$ and the payoffs for the 1PG. It appears that subjects who fully solved the 1PG are better at best responding to their own beliefs and therefore have a lower $\Delta z_i^{\ast }$ (Mann–Whitney U test p value $=0.001$ ). This is confirmed by a significantly negative correlation between 1PG payoffs and $\Delta z_i^{\ast }$ (Spearman $\rho =-0.531$ , p value $<0.001$ ). These results imply that better understanding of the structure of the guessing game improves the ability to best respond to own beliefs.

Result 5

Subjects with a better understanding of the structure of the one-player guessing game choose numbers closer to the best response of their beliefs in the two-player guessing game.

3.4.1 “What-if” beliefs

As there could be some influence of having played the 1PG on the beliefs in the 2PG, we asked 40 subjects to use 19 tokens to guess the choices of 19 subjects that had played the 2PG “a couple of weeks ago, without having previously played the 1PG”. We will refer to these distributions as “what-if” distributions, as opposed to the elicited distributions in the belief elicitation part of the experiment which we will refer to as “original” distributions.

Fig. 8 Distribution of “original” and “what-if” beliefs for the 2PG

Fig. 9 Payoff in the 1PG on the horizontal axis, and change in the mean between belief distributions ( $\Delta B_i$ ) on the vertical axis. Any value of $\Delta B_i$ above zero is a change in the mean away from the Nash Equilibrium