Guessing games, seemingly simple on the surface, offer a fascinating lens into human behavior and strategic decision-making. While you might not be diagnosing car troubles here at xentrydiagnosis.store, the underlying cognitive processes in a “Guess The Diagnosis Game” share surprising similarities with complex problem-solving scenarios. This article delves into an intriguing experiment that uses a unique guessing game to explore how individuals understand strategic interactions and form beliefs in competitive environments.
This research, involving 80 participants from diverse academic backgrounds, investigated behavior in two types of guessing games: a one-player guessing game (1PG) and a two-player guessing game (2PG). The study, conducted at the Experimental Economics Laboratory of the Technische Universität Berlin, aimed to understand how experience in a simpler game (the 1PG) influences strategic choices and understanding in a more complex one (the 2PG). Participants, primarily undergraduate students, were carefully selected to exclude those with prior experience in similar guessing game experiments.
Understanding the One-Player Guessing Game: A Solitary Diagnostic Challenge
The one-player guessing game (1PG) presents a unique cognitive puzzle. In this game, participants choose two numbers, aiming to minimize both their individual values and the difference between them. Figure 1 visually represents the choices made by participants in the 1PG.
Figure 1: Individual choices in the 1PG. Darker points indicate participants who correctly identified the optimal solution (0,0), showcasing their mastery of this “guess the diagnosis game” variant.
The scatter plot reveals a key finding: only about 31% of participants fully grasped the optimal strategy in the 1PG, which is to select zero for both numbers. This ability to “solve” the 1PG, meaning selecting (0,0), was used as a primary measure of understanding the game’s structure.
Key Finding 1: Limited Mastery of the One-Player Game
Result 1: A significant majority, approximately 69%, of participants did not fully understand the one-player guessing game and failed to identify the optimal solution.
Interestingly, the data showed a correlation: participants who chose numbers closer to each other also tended to choose numbers closer to zero. In the 1PG, improving one’s score involves either choosing numbers closer to zero, or narrowing the gap between the two chosen numbers. This suggests that even without fully solving the game, some participants intuitively grasped elements of the underlying payoff structure. Statistical analysis (Spearman correlation) confirmed a strong positive correlation between the average of the chosen numbers and their proximity to each other, indicating that higher scores in the 1PG reflected a partial understanding of the game’s mechanics. Therefore, the payoffs in the 1PG served as a secondary measure of understanding, complementing the primary measure of “solved” or “not solved.”
Navigating the Two-Player Guessing Game: Introducing Strategic Interaction
The research then moved to a two-player guessing game (2PG), introducing the element of strategic interaction. Figure 2 compares the distribution of choices in the 2PG between participants with and without prior experience in the 1PG.
Figure 2: Choice distributions in the 2PG. The left panel shows participants with 1PG experience, and the right panel shows those without, highlighting the impact of prior experience on strategic decisions in this “guess the diagnosis game” scenario.
The distribution of choices in the 2PG differed significantly from typical “first-timer” behavior in guessing games. For those with 1PG experience (left panel), a significant portion (50%) chose the Nash Equilibrium, with the distribution heavily skewed towards zero. The average choice was 13.47, and the median was 2. In contrast, participants without prior 1PG experience (right panel) exhibited a distribution shifted to the right, with a mean of 27.8 and a median of 26. While a notable percentage (28.75%) still played the Nash Equilibrium, the overall choices were significantly higher (Kolmogorov Smirnov test confirmed this statistical difference).
This difference in behavior suggests that prior experience with the 1PG influenced strategic thinking in the 2PG. This could be attributed to introspective learning from the 1PG, or a change in beliefs about the “experience level” of the participant pool in the 2PG. While belief shifts were found to be relatively small, the study leans towards introspective learning as the primary driver of the observed behavioral differences. Even though most participants didn’t fully solve the 1PG, the experience seemed to facilitate some level of strategic learning transferable to the 2PG.
Connecting the Games: How 1PG Understanding Predicts 2PG Strategy
To further explore the link between understanding in the 1PG and strategy in the 2PG, Figure 3 plots individual choices in the 2PG against their payoffs in the 1PG.
Figure 3: Correlation between 2PG choices (vertical axis) and 1PG payoffs (horizontal axis). Participants who excelled in the 1PG tended to choose lower numbers in the 2PG, indicating a transfer of strategic understanding in this “guess the diagnosis game” context.
Participants who had solved the 1PG (indicated by solid circles) predominantly chose zero in the 2PG (96% of them), and selected significantly lower numbers overall compared to those who hadn’t solved the 1PG (Mann-Whitney U test). Consistent with this, higher payoffs in the 1PG correlated with lower number choices in the 2PG (Spearman correlation).
Key Finding 2: 1PG Understanding Guides 2PG Strategy
Result 2: Participants who demonstrated a better grasp of the one-player guessing game by achieving higher payoffs tended to play numbers closer to the Nash Equilibrium in the two-player guessing game.
However, the study also questioned whether playing strictly according to the Nash Equilibrium was the optimal strategy in the 2PG. To investigate this, they calculated (bar{Pi }^{2PG}_i), representing the payoff each participant i would have received if they played against the average choice of all other participants.
Figure 4 examines the relationship between (bar{Pi }^{2PG}_i) and both 1PG payoffs (left panel) and 2PG choices (right panel).
Figure 4: Expected 2PG payoffs in relation to 1PG payoffs (left panel) and 2PG choices (right panel). The left panel’s quadratic fit suggests a complex relationship between game understanding and optimal strategy in this “guess the diagnosis game” scenario.
Intriguingly, participants who fully solved the 1PG did not achieve the highest (bar{Pi }^{2PG}_i). While they played the Nash Equilibrium (zero), payoffs were maximized by choosing a number closer to 9, as shown in the right panel. Overall, there was no significant difference in payoffs between those who solved the 1PG and those who didn’t (Mann-Whitney U test).
Analyzing 1PG payoffs as a secondary measure of understanding revealed a non-monotonic relationship with expected 2PG payoffs. Regression analysis supported a quadratic relationship, suggesting that while increased understanding initially leads to higher expected payoffs, this reverses at very high levels of understanding – indicating potential overthinking or overly sophisticated strategy in this context.
Key Finding 3: Non-Linearity of Understanding and 2PG Payoffs
Result 3: The connection between understanding the one-player guessing game and achieving higher payoffs in the two-player guessing game is not straightforward, following a non-monotonic pattern.
Belief Formation: Predicting Opponent Behavior in the “Guess the Diagnosis Game”
The study further investigated subjective beliefs. Participants were asked to distribute tokens to represent their beliefs about the choices of others in the 2PG. Figure 5 examines the relationship between 1PG payoffs and the accuracy of these beliefs.
Figure 5: Belief elicitation and 1PG payoffs. The left panel shows the correlation between 1PG payoff and correctly placed tokens. The right panel illustrates token distribution across strategy space, showing how participants with higher 1PG scores have more focused beliefs in this “guess the diagnosis game” study.
Participants who solved the 1PG placed a significantly larger number of tokens correctly (Mann-Whitney U test), confirmed by a strong correlation between 1PG payoff and the number of correctly placed tokens (Spearman correlation). Furthermore, those who solved the 1PG expected their counterparts to play numbers closer to the Nash Equilibrium, whereas those who didn’t solve the 1PG distributed their tokens more broadly across the strategy space (Mann-Whitney U test). The distance of tokens to the Nash Equilibrium also negatively correlated with 1PG payoffs (Spearman correlation).
To assess belief accuracy, Figure 6 plots the mean of each participant’s belief distribution against their 1PG payoff, and the absolute distance of individual mean beliefs to the average 2PG choice.
Figure 6: Belief accuracy and 1PG payoffs. The left panel compares mean belief value to 1PG payoff. The right panel shows the relationship between the absolute difference of mean beliefs from average 2PG choice and 1PG payoff, illustrating how those skilled in the 1PG are better at predicting group behavior in this “guess the diagnosis game” experiment.
The analysis revealed that subjects who solved the 1PG had a significantly smaller absolute difference between their mean beliefs and the average 2PG choice (Mann-Whitney U test). This was supported by a negative correlation between 1PG payoffs and this absolute difference (Spearman correlation).
Key Finding 4: Improved Belief Accuracy with 1PG Understanding
Result 4: Participants who demonstrated a better understanding of the one-player guessing game formed more accurate beliefs about their counterparts’ choices in the two-player guessing game.
Best Response to Beliefs: Strategic Consistency in Decision-Making
Finally, the study examined whether 2PG choices aligned with the participants’ stated beliefs. Figure 7 illustrates the relationship between the difference between actual 2PG choice and the optimal choice based on beliefs ((Delta z_i^*)) and 1PG payoffs.
Figure 7: Deviation from best response ((Delta z_i^*)) versus 1PG payoffs. Participants with higher 1PG scores demonstrated a smaller deviation, indicating a greater ability to act consistently with their beliefs in this “guess the diagnosis game” scenario.
Participants who solved the 1PG exhibited a significantly lower (Delta z_i^*) (Mann-Whitney U test), indicating they were better at choosing numbers closer to the best response given their beliefs. This was confirmed by a negative correlation between 1PG payoffs and (Delta z_i^*) (Spearman correlation).
Key Finding 5: Enhanced Best Response Ability with 1PG Understanding
Result 5: Participants with a better understanding of the one-player guessing game chose numbers that were closer to the optimal best response to their own beliefs in the two-player guessing game.
Exploring “What-If” Scenarios: Adapting Beliefs to Different Populations
To investigate the potential influence of 1PG experience on beliefs in the 2PG, a “what-if” scenario was explored. Participants were asked to predict the choices of individuals who played the 2PG without prior 1PG experience. Figure 8 compares the “original” belief distributions with these “what-if” distributions.
Figure 8: Distribution of “original” and “what-if” beliefs for the 2PG, illustrating how participants adjust their predictions based on the perceived experience level of the other players in this “guess the diagnosis game” study.
Visually, the differences appear subtle. However, statistical comparison revealed that both the mean and variance of individual distributions were significantly higher in the “what-if” scenario (Wilcoxon matched-pairs signed-ranks test). This suggests participants did adjust their beliefs based on the perceived experience of the population they were facing.
Figure 9 further examines this by plotting the difference in means between “what-if” and “original” distributions ((Delta B_i)) against 1PG payoffs.
Figure 9: Belief adjustment ((Delta B_i)) and 1PG payoffs. Positive values indicate a shift away from the Nash Equilibrium in the “what-if” beliefs. The figure explores whether those skilled in the 1PG are better at adjusting their predictions when facing less experienced players in this “guess the diagnosis game” experiment.
The data suggests that participants who solved the 1PG, when they adjusted their beliefs, tended to shift them away from the Nash Equilibrium – potentially a correct adjustment when facing a less experienced population. While statistical significance for the difference in distribution means ((Delta B_i)) between solvers and non-solvers of the 1PG was not found (Mann-Whitney U Test), a significant positive correlation between 1PG payoffs and (Delta B_i) (Pearson correlation) provides weak evidence for this.
Key Finding 6: Adaptive Belief Adjustment with 1PG Understanding (Weak Evidence)
Result 6: There is limited evidence suggesting that participants with a better understanding of the one-player guessing game may be better at adjusting their beliefs when anticipating the behavior of a less experienced population in the two-player guessing game.
Conclusion: Strategic Thinking Beyond the “Guess the Diagnosis Game”
This study provides valuable insights into strategic thinking and decision-making. While framed around a “guess the diagnosis game” concept for illustrative purposes, the core findings reveal fundamental aspects of how individuals learn in strategic environments, form beliefs about others, and adapt their strategies based on experience and understanding. The one-player guessing game, despite its simplicity, serves as a powerful predictor of behavior in more complex strategic interactions like the two-player guessing game. The research highlights the importance of cognitive skills and strategic learning in navigating even seemingly simple games, with implications that extend to more complex real-world scenarios beyond the laboratory.
