3.1 Normal form games and payoff matrices

Normal form games and payoff matrices are essential tools for analyzing strategic interactions. They provide a clear way to represent players, strategies, and payoffs in simultaneous move games, allowing us to visualize all possible outcomes.

Understanding these concepts is crucial for identifying Nash equilibria and dominant strategies. By examining payoff matrices, we can analyze strategic interdependence and determine best responses, key elements in predicting game outcomes and player behavior.

Normal Form Representation of Games

Representing Simultaneous Move Games

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• Normal form games represent strategic situations where players make decisions simultaneously without knowing the decisions of other players
• A normal form game specifies:
  • The players in the game
  • The strategies available to each player
  • The payoff received by each player for each combination of strategies that could be chosen by the players
• Normal form representation captures the key elements of a game: players, strategies, and payoffs

Using Payoff Matrices

• Payoff matrices are a convenient way to represent normal form games, listing the players, player strategies, and payoffs in a grid format
  • Player strategies are listed on the dimensions of the matrix
    • The row player's strategies are listed on the left
    • The column player's strategies are listed on the top
  • Each cell in the matrix shows the payoffs to the row and column player for the corresponding row and column strategies
  • By convention, payoffs are listed with the row player's payoff first, followed by the column player's payoff
• Payoff matrices provide a clear visual representation of the game, making it easier to analyze player incentives and outcomes (Prisoner's Dilemma, Battle of the Sexes)

Interpreting Payoff Matrices

Understanding Outcomes and Payoffs

• Each cell in a payoff matrix represents an outcome, a combination of strategies chosen by the players and the resulting payoffs
• To determine the payoff for a player from a particular outcome:
  • Identify the row corresponding to the row player's strategy
  • Identify the column corresponding to the column player's strategy
  • The cell at the intersection shows the payoffs to each player
• Payoffs are the values that each player receives at the outcome of the game based on the combination of strategies chosen
  • Payoffs are often represented as numerical values but can take other forms as long as they show the rank-ordering of player preferences (utility values, win/lose, years in prison)

Analyzing All Possible Outcomes

• The payoff matrix allows you to see all possible combinations of strategies and the resulting payoffs
• By examining the entire matrix, you can:
  • Identify the payoffs each player receives for each combination of strategies
  • Determine which outcomes are more or less desirable for each player
  • Analyze player incentives and likely strategies based on the payoff structure
• Seeing all potential outcomes helps determine the likely results of the game (Nash Equilibrium, dominant strategies)

Strategic Interdependence in Games

Optimal Strategies and Opponent Actions

• In simultaneous move games, the optimal strategy for a player depends on the strategy chosen by the other player(s)
  • This relationship between the best strategies of players is known as strategic interdependence
• Each player must anticipate the actions of their opponent(s) and choose their strategy based on how they expect their opponent(s) to play the game
  • Players consider questions like: "What is my opponent likely to do?" and "What is my best strategy given what I think my opponent will do?"

Analyzing Strategic Interdependence

• To analyze the strategic interdependence, examine how each player's payoffs change in response to changes in the other player's strategy choice
  • Look at a player's payoffs in each row (for the row player) or column (for the column player) to see how their payoffs change as the other player's strategy changes
  • If a player's best strategy depends on the other player's choice, then there is strategic interdependence
  • If a player has a single best strategy no matter what the other player does, then their optimal choice is independent of their opponent
• The presence or absence of strategic interdependence shapes the incentives and outcomes in the game (Matching Pennies vs. Prisoner's Dilemma)

Best Responses in Simultaneous Games

Defining Best Responses

• A player's best response is the strategy that gives them the highest payoff given the strategy chosen by the other player
• The best response is the optimal strategy choice for a player assuming their opponent's action is fixed
• Best responses indicate what each player should do to maximize their own payoff in light of their opponent's strategy

Finding Best Responses

• To find the best response, look at the payoffs in the matrix while holding the other player's strategy constant
  • For the row player, look across each row (holding the column player's strategy constant) to find the highest payoff
    • The row player's strategy corresponding to the highest payoff is their best response to that column strategy
  • For the column player, look down each column (holding the row player's strategy constant) to find the highest payoff
    • The column player's strategy corresponding to the highest payoff is their best response to that row strategy
• A player may have a single best response or multiple best responses to a particular strategy by their opponent

Best Responses and Nash Equilibrium

• If both players are simultaneously playing a best response to the other player's strategy, then the combination of strategies is a Nash equilibrium
• At a Nash equilibrium, no player can unilaterally improve their payoff by changing strategies
  • Each player is doing the best they can given the strategy of their opponent
• Finding best responses is a key step in identifying Nash equilibria in simultaneous move games (Prisoner's Dilemma, Battle of the Sexes)