3.2 Backward Induction and Strategic Planning

Backward induction is a powerful tool in game theory for solving sequential-move games. By working backward from the end of the game tree, players can determine optimal strategies and predict outcomes, considering the best responses of other players.

This approach helps players make informed decisions by anticipating future moves and assessing the credibility of threats and promises. Understanding backward induction is crucial for developing effective strategies in various real-world scenarios, from chess to business negotiations.

Backward Induction and Strategic Planning

Backward induction algorithm application

• Backward induction is an algorithm used to solve sequential-move games by working backward from the end of the game tree to determine the optimal strategy for each player
• Steps in the backward induction algorithm:
  1. Start at the end of the game tree and identify the optimal decision for the last player to move
  2. Move up the game tree to the second-to-last decision node and determine the optimal decision for the player at that node, assuming the last player will make their optimal choice
  3. Continue this process, moving up the game tree until reaching the initial decision node
• The resulting strategy profile determined by backward induction is the subgame perfect Nash equilibrium, a refinement of the Nash equilibrium that accounts for the credibility of threats and promises in sequential-move games
• Backward induction helps players make optimal decisions by anticipating future moves and considering the best responses of other players (chess, negotiations)

Game outcome prediction

• To predict the outcome of a sequential-move game, start at the terminal nodes of the game tree, which represent the possible outcomes of the game
• Assign payoffs to each player at the terminal nodes based on the game's payoff structure (monetary rewards, points)
• Work backward from the terminal nodes to determine the optimal decision for each player at their respective decision nodes
  • The optimal decision maximizes the player's payoff, assuming all subsequent players will also make optimal decisions
• By working backward and determining the optimal decision at each node, you can predict the game's outcome and the strategies players will employ
• Predicting game outcomes helps players make informed decisions and develop effective strategies (poker, business negotiations)

Anticipation in decision-making

• In sequential-move games, players must anticipate their opponents' future moves when making decisions because earlier decisions can influence the available options and payoffs later in the game
• Players use backward induction to reason about future moves and make decisions accordingly
  • By considering the optimal decisions of players who move later, a player can choose the best strategy at their current decision node
• The ability to anticipate future moves allows players to make credible threats or promises
  • A credible threat is a strategy that a player would actually carry out if the game reaches that point, as it is in their best interest to do so (threatening to cut prices in a market share battle)
  • A credible promise is a strategy that a player commits to, and it is in their best interest to follow through on that commitment (promising to cooperate in a repeated game)
• Anticipating future moves enables players to make strategic decisions and influence the game's outcome (product launches, international diplomacy)

Credibility in sequential games

• Credibility and commitment are essential concepts in sequential-move games that influence the strategies players choose and the game's outcome
• A threat or promise is credible if the player making it has an incentive to follow through on it
  • If a threat or promise is not credible, opponents may not believe it and act accordingly
• Commitment devices can be used to make threats or promises more credible by limiting one's own future choices, making it more likely that a player will follow through on a threat or promise
  • Examples of commitment devices include contracts, reputation, and irreversible actions (signing a binding agreement, publicly announcing a strategy)
• The credibility of threats and promises can alter the game's equilibrium outcome
  • If a threat or promise is credible, it may deter or encourage certain actions from other players, leading to a different outcome than if the threat or promise were not credible (deterring entry in a market, encouraging cooperation in a partnership)
• Credibility and commitment play a crucial role in shaping player behavior and determining the outcome of sequential-move games (labor negotiations, international treaties)