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Game Theory 101 (#6): Best Responses
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Game Theory 101 (#6): Best Responses

Introduction to Best Responses in Game Theory

In this section, William Spaniel introduces the concept of best responses in game theory and explains their significance in determining Nash equilibria.

Best Responses and Nash Equilibrium

  • A strategy is considered a best response if a player cannot gain more utility by switching to a different strategy.
  • A game is in Nash equilibrium when all players are playing best responses to what other players are doing.

Marking Best Responses

  • Best responses can be marked by isolating a single player's move and determining the optimal response for the other player(s).
  • Payoffs are compared for each possible outcome, and the highest payoff is marked as the best response.
  • This process is repeated for all strategies of each player to identify all best responses in the game.

Safety in Numbers Game

In this section, William Spaniel introduces the "Safety in Numbers" game, where two generals decide how many units to allocate for an upcoming battle. The outcome depends on the number of units allocated by each general.

Game Setup

  • Two generals simultaneously decide how many units they should allocate for battle.
  • Either side can unilaterally opt out of the battle by passing.
  • The side with more units wins, but if both sides have equal units or one side passes, it results in a draw.

Payoff Matrix

  • Each general has four strategies: pass, send 1 unit, send 2 units, or send 3 units.
  • There are seven possible outcomes based on the allocation of units.
  • If both sides pass, it's an automatic draw.
  • If both sides send an equal number of units, it's also a draw.
  • If one general sends more units than the other, that general wins.

Finding Pure Strategy Nash Equilibria

In this section, William Spaniel discusses the challenge of finding pure strategy Nash equilibria when there are numerous outcomes in a game. He introduces the concept of using best responses to identify Nash equilibria more efficiently.

Challenge of Large Games

  • When a game has many possible outcomes, it becomes time-consuming to analyze each individual outcome for profitable deviations.
  • The "Safety in Numbers" game has 16 different outcomes, but games can be even larger.
  • A more efficient method is needed to determine pure strategy Nash equilibria.

Marking Best Responses

  • Best responses can be used to identify pure strategy Nash equilibria.
  • By marking best responses, we can determine which strategies are optimal for each player without analyzing every single outcome.
  • This approach simplifies the process and saves time when dealing with large games.

Definition of Best Response and Nash Equilibrium

In this section, William Spaniel provides a formal definition of best response and explains how it relates to Nash equilibrium.

Definition of Best Response

  • A strategy is considered a best response if and only if a player cannot gain more utility by switching to a different strategy.
  • A game is in Nash equilibrium if and only if all players are playing best responses to what other players are doing.

Simplified Definition

  • The definition of Nash equilibrium is similar to what was previously discussed but now includes the term "best response."
  • The focus is on strategies that maximize utility given what other players are doing.

Marking Best Responses in the Game

In this section, William Spaniel demonstrates how to mark best responses in the "Safety in Numbers" game by considering each player's strategy and identifying the optimal response for the other player.

Marking Best Responses

  • Start by isolating one player's move and analyze the payoffs for all possible outcomes.
  • Identify the highest payoff as the best response for that player's strategy.
  • Repeat this process for all strategies of each player to mark all best responses in the game.

Multiple Best Responses

In this section, William Spaniel explains that there can be multiple best responses in a game and demonstrates how to mark them accordingly.

Multiple Best Responses

  • In some cases, there may be more than one best response for a particular strategy.
  • When marking best responses, it is important to identify all optimal outcomes for each player's strategy.
  • These multiple best responses are marked accordingly to reflect their significance in determining Nash equilibria.

Marking Player 1's Best Responses

In this section, the speaker discusses marking all of Player 1's best responses to Player 2's moves.

Marking Player 1's Best Responses

  • Player 2 is indifferent between all of Player 1's strategies when Player 1 passes.
  • When Player 1 plays one unit, Player 2 can get zero or one payoff.
  • When Player 1 selects two units, Player 2 can get negative one or one payoff.
  • When Player 1 sends out three units, Player 2 can get negative one or zero payoff.

Nash Equilibrium and Best Responses

This section explains the concept of Nash equilibrium and how it relates to best responses in a game.

Nash Equilibrium and Best Responses

  • A game is in a Nash equilibrium if all players are playing best responses to what the other players are doing.
  • Boxes with two asterisks represent mutual best responses between players.
  • Four boxes have two asterisks each, indicating pure strategy Nash equilibria.
  • The rest of the outcomes do not have mutual best responses and therefore cannot be Nash equilibria.

Efficiency of Finding Pure Strategy Nash Equilibria

This section highlights the efficiency of using the best response method to find pure strategy Nash equilibria.

Efficiency of Finding Pure Strategy Nash Equilibria

  • Using the best response method is more efficient than considering all possible outcomes.
  • By marking the best responses, only four different strategies need to be considered instead of sixteen different situations.
  • It is recommended to mark the best responses in more complicated games to identify pure strategy Nash equilibria efficiently.
Video description

Game Theory 101: The Complete Textbook on Amazon: https://www.amazon.com/Game-Theory-101-Complete-Textbook/dp/1492728152/ http://gametheory101.com/courses/game-theory-101/ Finding pure strategy Nash equilibria was easy when there were only four outcomes. But if there are a lot more outcomes, say 16, going through each of them individually would be far too time consuming. This lesson shows how to find pure strategy Nash equilibria using the best responses method. To illustrate the concept, we use the safety in numbers game. Two generals decide whether to pass on fighting a battle or sending one, two, or three units. If at least one general decides not to fight or the generals send the same number of units, the game ends in a draw. Otherwise, the general with the most number of units wins.