1. Strategizing with AI: Insights from a Beauty Contest Experiment (joint with Iuliia Alekseenko, Sofia Paklina and Petr Parshakov; Revise and Resubmit, JEBO) (pdf)
A Keynesian beauty contest is a wide class of games of guessing the most popular strategy among other players. In particular, guessing a fraction of a mean of numbers chosen by all players is a classic behavioral experiment designed to test level-k reasoning patterns among various groups of people. The previous literature reveals that the sophistication level of opponents is an important factor affecting the outcome of the game. Smarter decision makers choose strategies that are closer to theoretical Nash equilibrium and demonstrate faster convergence to equilibrium in iterated contests with information revelation. In the level-k reasoning framework, the Nash equilibrium is played only by infinitely advanced players. We run a series of virtual experiments with an AI player, GPT-4, who plays against various groups of players. We test how advanced is this learning language model compared to human players by replicating some of the classic experiments. It is shown that GPT-4 takes into account the opponents' level of sophistication and adapts by changing the strategy. However, the transformation of the particular values of parameters to output data does not necessarily respect the comparative statics of the model. Lasso regression analysis revealed a closer alignment of AI-generated guesses to strategic thinking compared to human participants. Our results contribute to the discussion on the accuracy of modeling human economic agents by artificial intelligence.
2. Strategic Broadcasting of a Championship: to Earn More from Ads or to Incentivize Subscription? (joint with Igor Karpov; Revise and Resubmit, International Journal of Sport Finance) (pdf)
Consider a company that owns two sports TV channels: a popular free-to-air channel and a narrow-profile pay one. The company bought the broadcasting rights for a round-robin championship. When developing the broadcasting strategy, the company's management faces the trade-off between inducing viewers to subscribe to the pay channel and getting revenues from advertisements on the free-to-air channel. Motivated by an anecdotal case of a significant drop in the percentage of broadcasts of best teams among all broadcasts on a free-to-air channel, we propose a theoretical model of broadcasting the championship. We demonstrate that a wide range of strategies can be optimal depending on the market parameters. Whereas the intermediate strategy of broadcasting just weak matches (instead of broadcasting top matches or broadcasting nothing) on the free-to-air channel seems puzzling, we show formally that it can be fully rational for some market configurations. Therefore, we rationalize the puzzling strategy used by the company's management in the motivational example.
3. Implicit Centipedes (joint with Daniil Starikov and Gleb Vasiliev)
The centipede game illustrates a strategic interaction in which real-world players often deviate from the subgame perfect Nash equilibrium. In previous laboratory experiments, participants typically observed the full game tree, including payoff structures. Even under these relatively simple conditions, cognitive capacity has been linked to adherence to the equilibrium path. However, the explicit presentation of the game may significantly influence this outcome. Consider instead a more complex, implicitly defined problem that effectively constitutes a centipede game. In such cases, more advanced players may perform better by inferring the underlying structure of the game. Conversely, if the game is too complex to be solved — even by highly skilled players — then the resulting interaction may diverge significantly from equilibrium predictions, leading to unpredictable outcomes. To explore the dynamics of implicitly formulated games, we examine an antagonistic variant of the centipede game played by professional athletes—specifically, Formula 1 drivers and their teams—during races. When two drivers are in close proximity, they engage in a strategic contest with the goal of finishing ahead of one another. Each lap presents a decision point: the leading driver and the pursuer sequentially decide whether to pit and change tires. Analyzing intra-race data from multiple Formula 1 seasons, we demonstrate that this scenario constitutes a complex, centipede-like game. Although the drivers do not observe the explicit game tree, our findings reveal that more skilled drivers and teams adopt strategies that more closely align with theoretical equilibrium predictions, despite the game's implicit nature.
4. Antagonistic sequential games with ties between players with limited search capacity
I consider an important class of antagonistic sequential games of value 0 with ties, where players do not have enough memory capacity to solve the game using backward induction. Checkers and supposedly chess belong to this class. In such games the level of human players is associated with the number and severity of mistakes (deviations from the subgame perfect equilibrium). One of the most popular ways to predict an outcome of such games is based on the paired comparison model. I show formally that for this class of games a predictive model that matches the empirical evidence, cannot be obtained in a paired comparison framework: two types of the desirable monotonicity lead to incompatibility. A relaxation of monotonicity properties that allows a solution to be found is also proposed in the paper.
A Keynesian beauty contest is a wide class of games of guessing the most popular strategy among other players. In particular, guessing a fraction of a mean of numbers chosen by all players is a classic behavioral experiment designed to test level-k reasoning patterns among various groups of people. The previous literature reveals that the sophistication level of opponents is an important factor affecting the outcome of the game. Smarter decision makers choose strategies that are closer to theoretical Nash equilibrium and demonstrate faster convergence to equilibrium in iterated contests with information revelation. In the level-k reasoning framework, the Nash equilibrium is played only by infinitely advanced players. We run a series of virtual experiments with an AI player, GPT-4, who plays against various groups of players. We test how advanced is this learning language model compared to human players by replicating some of the classic experiments. It is shown that GPT-4 takes into account the opponents' level of sophistication and adapts by changing the strategy. However, the transformation of the particular values of parameters to output data does not necessarily respect the comparative statics of the model. Lasso regression analysis revealed a closer alignment of AI-generated guesses to strategic thinking compared to human participants. Our results contribute to the discussion on the accuracy of modeling human economic agents by artificial intelligence.
2. Strategic Broadcasting of a Championship: to Earn More from Ads or to Incentivize Subscription? (joint with Igor Karpov; Revise and Resubmit, International Journal of Sport Finance) (pdf)
Consider a company that owns two sports TV channels: a popular free-to-air channel and a narrow-profile pay one. The company bought the broadcasting rights for a round-robin championship. When developing the broadcasting strategy, the company's management faces the trade-off between inducing viewers to subscribe to the pay channel and getting revenues from advertisements on the free-to-air channel. Motivated by an anecdotal case of a significant drop in the percentage of broadcasts of best teams among all broadcasts on a free-to-air channel, we propose a theoretical model of broadcasting the championship. We demonstrate that a wide range of strategies can be optimal depending on the market parameters. Whereas the intermediate strategy of broadcasting just weak matches (instead of broadcasting top matches or broadcasting nothing) on the free-to-air channel seems puzzling, we show formally that it can be fully rational for some market configurations. Therefore, we rationalize the puzzling strategy used by the company's management in the motivational example.
3. Implicit Centipedes (joint with Daniil Starikov and Gleb Vasiliev)
The centipede game illustrates a strategic interaction in which real-world players often deviate from the subgame perfect Nash equilibrium. In previous laboratory experiments, participants typically observed the full game tree, including payoff structures. Even under these relatively simple conditions, cognitive capacity has been linked to adherence to the equilibrium path. However, the explicit presentation of the game may significantly influence this outcome. Consider instead a more complex, implicitly defined problem that effectively constitutes a centipede game. In such cases, more advanced players may perform better by inferring the underlying structure of the game. Conversely, if the game is too complex to be solved — even by highly skilled players — then the resulting interaction may diverge significantly from equilibrium predictions, leading to unpredictable outcomes. To explore the dynamics of implicitly formulated games, we examine an antagonistic variant of the centipede game played by professional athletes—specifically, Formula 1 drivers and their teams—during races. When two drivers are in close proximity, they engage in a strategic contest with the goal of finishing ahead of one another. Each lap presents a decision point: the leading driver and the pursuer sequentially decide whether to pit and change tires. Analyzing intra-race data from multiple Formula 1 seasons, we demonstrate that this scenario constitutes a complex, centipede-like game. Although the drivers do not observe the explicit game tree, our findings reveal that more skilled drivers and teams adopt strategies that more closely align with theoretical equilibrium predictions, despite the game's implicit nature.
4. Antagonistic sequential games with ties between players with limited search capacity
I consider an important class of antagonistic sequential games of value 0 with ties, where players do not have enough memory capacity to solve the game using backward induction. Checkers and supposedly chess belong to this class. In such games the level of human players is associated with the number and severity of mistakes (deviations from the subgame perfect equilibrium). One of the most popular ways to predict an outcome of such games is based on the paired comparison model. I show formally that for this class of games a predictive model that matches the empirical evidence, cannot be obtained in a paired comparison framework: two types of the desirable monotonicity lead to incompatibility. A relaxation of monotonicity properties that allows a solution to be found is also proposed in the paper.
Economics, Game Theory, Computer Science, Sports Studies
Dmitry Dagaev
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