Do rationalizable strategies matter?
(Analysis of the Price War Game experiment)
Vladimír Gazda and Michal Páleník and Experiment Participants
Research topic
The popular statement on the existence of unique equilibrium prices as the consequence of the perfect competition assumptions (Walras, 2013) has been contradicted many times. The seminal paper by (Hotelling, 1929) explained the pricing and localization factors of selling homogeneous and heterogeneous goods. Empirical research has documented that prices can change significantly in different places [see (Janský & Kolcunová, 2017) and (Aten & D’Souza, 2008)]]. On the other hand, (Samuelson, 1954) invented the Iceber model, while (Cassel, 1918) proposed the Law of One Price. Both clearly explained the price differences as the results of the localization factors.
The presented research explains the price differences using the basic tools of Game theory when the Consumers-Sellers game has been reduced to a 2-Sellers price game. Sellers and Consumers are located in 2D space. Consumers decide which Seller to go shopping at based on the calculated costs of purchasing one piece of goods.
Model
Assumptions
- there is no agreement between the Sellers.
- Seller 1 and 2 choose prices \(p_1\) and \(p_2\) from \(\{1, 2, \dots, 6\}\).
- Consumers make deterministic decisions based on calculating the buying costs. These costs consist of two parts:
- the unit price of the good, and
- the traveling costs needed to reach a seller’s shop. If consumers have equal buying costs when visiting any seller, they will make random decisions.
Model core
Deterministic decision of the Consumers enables reduction of the complexity of the model. As the strategic Consumers’ decisions do not play any role, we consider just two players (Seller 1 and Seller 2) which utilize their price strategy spaces \(\{1,2, \dots, 6\}\) in the Price War Game.
Assume a situation with two Sellers and four Consumers. The following figure shows their location in a space.
Spatial layout of Sellers and Consumers
The second Seller’s location is clearly advantageous, especially when considering the distances of the Consumers to the shops. Seller S2 will attract the three or four nearest customers. This raises scientific questions about the price policy of both Sellers:
- Does the price policy of both Sellers lead to a stable equilibrium?
- Do the resulting prices converge to their maxima (i.e. 6)?
- Will the second Seller push its competitor out of the market (i.e. will the price pressure enforce the first Seller of setting prices to the minimal levels)?
In the following, we assume the travelling costs as given in the Table below:
| S1 | S2 | |
|---|---|---|
| C1 | 1 | 3 |
| C2 | 3 | 1 |
| C3 | 3 | 2 |
| C4 | 4 | 2 |
The reduction of the price war game’s complexity consequently allows for the consideration of just two players (Seller 1 - S1 and Seller 2 - S2) and their strategy space: \(\{1, 2, ..., 6\}\). The expected payoffs depend on the strategy decision of both of them as given in the following Expected payoffs matrix:
| S1 \ S2 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 \ 3 | 1.5* \ 5* | 3 \ 3 | 4 \ 0 | 4 \ 0 | 4 \ 0 |
| 2 | 2* \ 3 | 2 \ 6 | 3 \ 7.5* | 6* \ 4 | 8 \ 0 | 8 \ 0 |
| 3 | 1.5 \ 3.5 | 3* \ 6 | 3 \ 9 | 4.5 \ 10* | 9* \ 5 | 12* \ 0 |
| 4 | 0 \ 4 | 2 \ 7 | 4* \ 9 | 4 \ 12 | 6 \ 12.5* | 12^* \ 6^ |
| 5 | 0 \ 4 | 0 \ 8 | 2.5 \ 10.5 | 5 \ 12 | 5 \ 15* | 7.5^ \ 15^* |
| 6 | 0 \ 4 | 0 \ 8 | 0 \ 12 | 3 \ 14 | 6 \ 15 | 6^ \ 18^* |
| Asterisks (*) denote Best responses. Carets (^) indicate Pareto optimal outcomes. |
Eyeballing the matrix, the following is obvious:
- strategy pairs \([4,6]\), \([5,6]\), and \([6,6]\) are Pareto optimal;
- there is no Nash equilibrium
- the price sets \({2,3,4}\) of Seller 1 and \({3,4,5}\) of Seller 2 are rationalizable strategies, as the other strategies are Never best-response strategies
Experiment
Fifteen students and one assistant professor participated in the experiment. The experiment consisted of eight rounds, as was known to the players before the experiment starts. Each participant had sufficient time to thoroughly study the matrix of expected payoffs. Half the students were assigned the role of Seller S1 (row player), and the other half were assigned Seller S2 (column player). In each round, the Sellers were randomly re-matched, while keeping the role of the particular Seller during the whole experiment. The experiment was run in the experimental portal of the Virginia university called VeconLab Game.
Experiment results
The experiment participant did not have any theoretical knowledge on the Best response strategies as well as the Rationalizable strategies. That is, why we consider their behaviour higly intuitive, while with the hope on the learning effect. Convergence towards the sets of the Rationalizable strategies was expected. The following eight figures describe the dynamics of the experimental results.
Results of the 1st round: Frequencies of the strategy profiles (\([p_1,p_2]\)) within the racionalizable strategies pairs is 6 against 2 profiles which were out of the rationalizable strategies pairs area (within the red border in the figure). On the other side, if speaking about individual strategies, 13 strategies were racionalizable and just 3 were the Never-best responses (= no Rationalizable strategy).
Results of the 2nd round: Frequencies of the strategy profiles (\([p_1,p_2]\)) within the racionalizable strategies pairs is 4 against 4 profiles which were out of the rationalizable strategies pairs area. On the other side, if speaking about individual strategies, 12 strategies were racionalizable and just 4 were the Never-best responses (= no Rationalizable strategy).
Results of the 3rd round: Frequencies of the strategy profiles (\([p_1,p_2]\)) within the racionalizable strategies pairs is 5 against 3 profiles which were out of the rationalizable strategies pairs area. On the other side, if speaking about individual strategies, 13 strategies were racionalizable and just 3 were the Never-best responses (= no Rationalizable strategy).
Results of the 4th round: Frequencies of the strategy profiles (\([p_1,p_2]\)) within the racionalizable strategies pairs is 6 against 2 profiles which were out of the rationalizable strategies pairs area. On the other side, if speaking about individual strategies, 14 strategies were racionalizable and just 2 were the Never-best responses (= no Rationalizable strategy).
Results of the 5th round: Frequencies of the strategy profiles (\([p_1,p_2]\)) within the racionalizable strategies pairs is 7 against 1 profiles which were out of the area of the rationalizable strategies pairs. On the other side, if speaking about individual strategies, 15 strategies were racionalizable and just 1 were the Never-best responses (= no Rationalizable strategy).
Results of the 6th round: Frequencies of the strategy profiles (\([p_1,p_2]\)) within the racionalizable strategies pairs is 7 against 1 profiles which were out of the area of the rationalizable strategies pairs. On the other side, if speaking about individual strategies, 15 strategies were racionalizable and just 1 were the Never-best responses (= no Rationalizable strategy).
Results of the 7th round: Frequencies of the strategy profiles (\([p_1,p_2]\)) within the racionalizable strategies pairs is 4 against 4 profiles which were out of the rationalizable strategies pairs area. On the other side, if speaking about individual strategies, 12 strategies were racionalizable and just 4 were the Never-best responses (= no Rationalizable strategy).
Results of the 8th round: Frequencies of the strategy profiles (\([p_1,p_2]\)) within the racionalizable strategies pairs is 7 against 1 profiles which were out of the area of the rationalizable strategies pairs. On the other side, if speaking about individual strategies, 14 strategies were racionalizable and just 2 were the Never-best responses (= no Rationalizable strategy).
Is there any learning effect? (Linear regression analysis)
The section analyzes the improvement of the Sellers decisions towards paying exclusively racionalizable profiles. If the strategy profile is not Racionalizable, then we consider it to be faulty. The regression explaining the number of faults on the round number, expresses the following analysis. We expect the learning effects should decrease the numebr of faulty profiles with the increasing rounds.
Analysis of the Strategy profiles
| Term | Estimate | Std. Error | t value | p value | Signif. |
|---|---|---|---|---|---|
| (Intercept) | 3.000 | 1.023 | 2.934 | 0.026 |
|
| round | -0.167 | 0.202 | -0.823 | 0.442 |
Residual Std. Error: 1.312
Multiple R-squared: 0.101
Adjusted R-squared: -0.048
F-statistic: 0.677 on 1 and NA DF, p-value: 0.442
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
The regression results conclude that, however, the slope of the regression line negative, it is not statistically signifficant, despite our expectations. The reason can be the small number of the degrees of freedom, but also the expectations of the final round in the round before the last, in which the players played quite irrationaly. We assume, the players wanted to enforce their oponents to play in a willing manner (expectation of the step-out effects).
Discussion
The experiment showed that, over repeated rounds, participants increasingly selected rationalizable strategies, even without formal training in game theory. This suggests intuitive learning and adaptation. However, the regression analysis did not find a statistically significant trend, likely due to the limited number of rounds and possible strategic experimentation near the end. The absence of a Nash equilibrium may have encouraged players to rely on heuristics rather than fixed strategies.
The study aimed to answer three main questions: (1) whether a stable equilibrium would emerge, (2) whether prices would converge to their maximum values, and (3) whether the better-positioned seller would push out the competitor. The results indicate that no stable equilibrium formed, consistent with the theoretical absence of a Nash equilibrium. Price strategies fluctuated and did not consistently converge to the upper bound. Lastly, although Seller 2 had a spatial advantage, this did not translate into clear market dominance across rounds, suggesting positional advantage alone was insufficient for exclusion.
An interesting direction for future research would be to analyze the dynamics of strategic responses across rounds. Specifically, one could investigate whether a participant’s current strategy depends on their opponent’s previous action. This line of inquiry could be formalized using Markovian modeling, where transition probabilities capture how players condition their strategies on past plays. Such an approach may offer deeper insight into the micro-level learning and adaptation mechanisms that shape the evolution of strategic behavior over time.
Conclusion
Rationalizable strategies proved useful in interpreting player behavior in the price war game. While not perfectly predictive, they reflected a general trend toward more rational decisions over time. Further experiments with larger samples, more rounds, and dynamic modeling—such as Markov-based cause-consequence analysis—could provide a richer understanding of strategy development and behavioral adaptation.