How do you handle dynamic and sequential games with changing payoffs and strategies over time?
Game theory is a powerful tool for analyzing strategic interactions among rational agents. However, many real-world situations are not static and one-shot, but dynamic and sequential, where the payoffs and strategies of the players change over time. How do you handle such complex scenarios with game theory? In this article, we will explore some key concepts and methods for dealing with dynamic and sequential games, such as backward induction, subgame perfect equilibrium, repeated games, and stochastic games.
Backward induction is a technique for finding the optimal strategy in a sequential game, where the players move one after another. It involves starting from the end of the game and working backwards, eliminating the strategies that are not optimal for each player at each stage. For example, in a game of chess, you can use backward induction to find the best move by considering all the possible responses of your opponent and choosing the one that maximizes your payoff. Backward induction assumes that the players have perfect information and rationality, and that the game has a finite horizon.
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Adam DeJans Jr.
Decision Science Leader @ Toyota | Drives Billion-Dollar Decisions | Optimization Strategist for Business Excellence | Author
Data scientists working on predictive modeling could use backwards induction to iterate over different model parameters and structures to find the most predictive model, essentially working backward to find the most robust predictive framework.
Subgame perfect equilibrium is a refinement of the Nash equilibrium concept, which applies to dynamic and sequential games. A Nash equilibrium is a set of strategies where no player can benefit by deviating unilaterally, given the other players' strategies. However, a Nash equilibrium may not be credible or consistent in a sequential game, where some strategies may involve empty threats or promises that are not fulfilled in later stages. A subgame perfect equilibrium is a Nash equilibrium that is also optimal for every subgame of the original game. A subgame is a smaller game that consists of some or all of the moves and payoffs of the original game. A subgame perfect equilibrium can be found by using backward induction.
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Robert I.
Sales and Marketing Leader | Go-to-Market Strategy | Specializing in eCommerce and Omnichannel | ex-Chobani | Angel Investor
In dynamic and sequential games, Subgame Perfect Equilibrium refines the Nash Equilibrium concept, ensuring strategies remain credible over time. Unlike Nash, where strategies might involve non-credible threats or promises: Credibility: Each stage's strategy must be optimal, ruling out empty threats. Backward Induction: Start from the game's end and work backwards to determine optimal strategies at each point. This approach ensures consistency and credibility in evolving game scenarios, vital for complex decision-making.
Repeated games are a type of dynamic game where the same players play the same game multiple times, with the possibility of observing the previous outcomes and adjusting their strategies accordingly. Repeated games can have different forms, such as finite or infinite horizon, perfect or imperfect monitoring, and discounting or no discounting. Repeated games allow for the emergence of cooperation, reputation, and punishment among the players, which can affect the equilibrium outcomes. For example, in the repeated prisoner's dilemma, where two players can choose to cooperate or defect, the players may sustain cooperation by using strategies such as tit-for-tat or grim trigger, which reward or punish the other player based on their past behavior.
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Adam DeJans Jr.
Decision Science Leader @ Toyota | Drives Billion-Dollar Decisions | Optimization Strategist for Business Excellence | Author
To improve the paragraph, we might consider: 1. Briefly defining what a “dynamic game” is to offer readers a foundational understanding. 2 Providing a simple explanation or examples for terms such as “finite or infinite horizon,” “perfect or imperfect monitoring,” and “discounting or no discounting.”
Stochastic games are a generalization of repeated games, where the payoffs and the transition probabilities between the states of the game depend on the actions of the players. Stochastic games can capture the uncertainty and randomness in dynamic and sequential games, as well as the feedback effects of the players' decisions on the future state of the game. Stochastic games can have different characteristics, such as zero-sum or non-zero-sum, cooperative or non-cooperative, and discrete or continuous. Stochastic games can be solved by using methods such as dynamic programming, Markov decision processes, or reinforcement learning.
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Ava Brill
In economics, stochastic games can model market dynamics affected by policies and external factors. In international relations, they can simulate diplomatic interactions where countries adjust their strategies based on others' actions. They're also instrumental in AI, where agents learn to navigate complex environments with changing rules and objectives. Their ability to model complex interactions and adapt to changing conditions makes them invaluable in fields ranging from economics to artificial intelligence, providing insights into how to navigate and make decisions in a world where change is the only constant.
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