5.1 Backward induction and sequential rationality
3 min read•august 7, 2024
and are key concepts in solving sequential games. They help players figure out the best moves by thinking through the game from end to start, assuming everyone acts rationally at each step.
These ideas are crucial for understanding in sequential games. They ensure players make smart choices throughout, not just at the beginning, based on what they know and expect others to do.
Backward Induction and Sequential Rationality
Solving Games through Backward Induction
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- Backward induction is a process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions
- Involves analyzing a from the end nodes back to the root node to determine the for each player
- Assumes that all players are rational and will make the best decision at each stage, given the actions of the other players
- Leads to the concept of , where each player's strategy is optimal given the strategies of the other players and the player's position in the game tree
Sequential Rationality and Optimal Strategies
- Sequential rationality requires that a player's strategy must be optimal at every point in the game, not just at the beginning
- Means that each player's strategy must be the best response to the other players' strategies, given the information available at each decision point
- Optimal strategy is a complete plan of action that specifies what a player will do at each decision point, contingent on what has happened previously in the game
- Ensures that players make the best decisions possible at each stage of the game, based on their beliefs about the other players' strategies and the information available to them
Extensive Form Games
Game Trees and Decision Nodes
- Extensive form games represent the sequential structure of a game, showing the order in which players make moves and the information they have when making decisions
- Game trees are graphical representations of extensive form games, with nodes representing decision points and edges representing actions or moves
- Decision nodes are points in the game tree where a player must make a choice among a set of possible actions
- Each path through the game tree represents a possible sequence of moves by the players, leading to a terminal node that specifies the payoffs for each player
Information Sets in Extensive Form Games
- Information sets are collections of decision nodes that a player cannot distinguish between when making a move
- Represent situations where a player has incomplete information about the previous moves of other players
- All nodes within an must have the same set of possible actions for the player making the decision
- Help to capture the idea of in a game, where players may not have complete knowledge of the game's history at every decision point
Information in Games
Perfect and Imperfect Information
- games are those in which each player, when making any decision, is perfectly informed of all the events that have previously occurred (chess, Go)
- Allows players to observe the entire history of the game and make decisions based on complete knowledge of the game state
- Imperfect information games are those in which players do not have complete knowledge of the previous moves or actions of other players (poker, bridge)
- Players must make decisions based on incomplete or uncertain information about the game's history or the strategies of other players
- Leads to more complex strategic considerations, as players must reason about the possible beliefs and strategies of their opponents based on the limited information available to them
Key Terms to Review (11)
Backward induction: Backward induction is a method used in game theory to determine optimal strategies by analyzing a game from the end to the beginning. It involves looking at the last possible moves of players and determining their best responses, then moving sequentially backward through the game tree to deduce the optimal actions of earlier moves. This technique is particularly relevant in analyzing strategic interactions in sequential games and helps in identifying subgame perfect equilibria.
Decision node: A decision node is a point in a decision tree or game tree where a player must make a choice among multiple options. These nodes are crucial for understanding the strategic interactions between players, as they represent the moments when decisions are made that affect future outcomes. The choices made at decision nodes can lead to different branches in the game tree, which are essential for analyzing strategies through concepts like backward induction and sequential rationality.
Extensive form game: An extensive form game is a representation of a strategic situation that allows players to make decisions at various points in time, depicted through a tree-like structure that illustrates the sequence of moves, choices, and potential outcomes. This format helps analyze strategies in situations where timing and order of moves matter, connecting key concepts like backward induction, sequential rationality, and subgame perfect equilibrium, while also illustrating credible threats and promises as well as the iterative elimination of dominated strategies.
Game Tree: A game tree is a graphical representation of a strategic game that illustrates the possible moves and outcomes from each player's decisions over time. It organizes the game's structure by showing the sequence of actions, including branching points for choices made by players, which leads to various terminal nodes representing outcomes. Game trees are essential for understanding how players can strategize in extensive form games and are particularly useful in applying concepts like backward induction to identify optimal strategies.
Imperfect Information: Imperfect information refers to a situation in a game where at least one player lacks complete knowledge about other players' actions, choices, or types. This lack of information can lead to uncertainty and strategic decision-making challenges, as players must make choices based on incomplete data. In this context, it impacts how players form beliefs, create strategies, and how the structure of games is understood, particularly in relation to decision-making processes and the overall framework of gameplay.
Information Set: An information set is a collection of decision nodes in a game where a player cannot distinguish between them, meaning the player does not know which node they are at when making a decision. This concept is crucial in understanding how players make strategic choices under uncertainty and helps to illustrate the differences between normal and extensive form representations of games, sequential decision-making processes, and the formation of beliefs in dynamic environments.
Optimal Strategy: An optimal strategy is a predetermined plan or course of action that yields the best possible outcome for a player, given the strategies of other players in a game. This concept is essential in decision-making processes and involves anticipating the actions of others to make the most effective choices. The optimal strategy is closely linked to backward induction and sequential rationality, as it often requires evaluating future outcomes based on present decisions.
Perfect Information: Perfect information refers to a situation in a game where all players are fully aware of all the actions that have taken place prior to their turn. This means every player knows the history of moves and choices made by others, allowing them to make informed decisions. This concept is essential for understanding the structure of extensive form games, where game trees represent the sequential nature of decision-making, and it plays a significant role in analyzing strategies through backward induction.
Rollback equilibrium: Rollback equilibrium is a solution concept in game theory that arises from backward induction, where players make optimal decisions at each stage of a game, considering the future actions of other players. This approach ensures that players' strategies are sequentially rational, meaning that their current choices are the best responses to future strategies. By anticipating how opponents will react at later stages, players can develop strategies that lead to an overall optimal outcome.
Sequential Rationality: Sequential rationality is a concept in game theory that describes a player's strategy as being optimal at every stage of a decision-making process, considering the future actions of other players. It implies that players make decisions not only based on the current situation but also take into account how their choices will affect subsequent moves in the game. This concept connects closely with the ideas of backward induction and how beliefs are updated in response to others' actions within strategic interactions.
Subgame Perfect Equilibrium: Subgame perfect equilibrium is a refinement of Nash equilibrium applicable to dynamic games where players make decisions at various stages. It requires that players' strategies constitute a Nash equilibrium in every subgame of the original game, ensuring that the strategies are credible and optimal, even if the game is played out from any point along the decision path.