Lesson 12: Individual Strategic Interactions

12.1 Learning Objectives

• Understand simultaneous games.

• Find the Nash equilibrium of the game.

• Understand Dominant strategy and how a dominant strategy does not imply Nash equilibrium.

• Learn that equilibrium exists while the presence of Nash equilibrium provides evidence of a dominant strategy.

• Understand sequential games.

• Solve sequential games.

12.2 Introduction

In this chapter, we will learn about simultaneous as well as sequential games in detail. Simultaneous games are those in which players have no knowledge about the actions chosen by other players, leading to imperfect information. Players are trying to predict the choices of other players while those players are simultaneously trying to predict theirs. There are two types of strategies players can choose: pure strategy and mixed strategy. This section will discuss pure strategy in detail. In a sequential game, one player moves first and gains a first-mover advantage. We will solve this type of game using backward induction.

12.3 Simultaneous Game

In a simultaneous game, each player makes a decision at the same time without knowing the choices of the other players, resulting in imperfect information. For example:

• In an auction, all bidders make their bids simultaneously without knowledge of others' bids.

• In the game of rock-paper-scissors, players have no information about the other players' actions.

A discrete strategy is defined as having a finite number of pure strategies. For instance, player 1 chooses between heads and tails. Strategies can be represented in a payoff matrix or game table, which must align dimensions with the number of players. The game can also be visualized in an extensive form.

Example of Payoff Matrix

Explanation of Payoff Matrix

In the example, player 1 has two choices (Top and Bottom), and player 2 has two choices (Left and Right). Outcomes are generated when specific row and column choices are made. For instance:

• If player 1 chooses 'Bottom' and player 2 chooses 'Left', the payoff is (4,2) where 4 is player 1's payoff and 2 is player 2's payoff.

• Similarly, if player 1 chooses 'Top' and player 2 chooses 'Right', the payoff is (3,5).

12.3.1 Nash Equilibrium

In the provided payoff matrix, given that player 1 chooses Top, player 2's best response is Right (5 > 3). Conversely, given player 1 chooses Bottom, player 2's best response is Left (2 > 1).

Definition of Nash Equilibrium: In Nash equilibrium, each player aims to maximize their payoff considering the choice of the other player. It represents a strategic choice from which no player has an incentive to deviate once set.

Verifying if (Top, Right) is Nash equilibrium:

• If player 1 chooses Top, player 2 should pick Right (5 vs. 3).

• If player 2 chooses Right, player 1's best option shifts to Bottom (5 vs. 3). Thus, (Top, Right) is not Nash equilibrium as it incentivizes player 1 to deviate.

Finding a Nash Equilibrium: (Bottom, Left) is a Nash equilibrium as both players have no incentive to deviate:

• For player 1, after player 2 chooses Left, the optimal choice remains Bottom (4 > 2).

• For player 2, after player 1 takes Bottom, their best response is Left (2 > 1).

12.3.2 Underline Best Response Payoff

To determine the Nash equilibrium:

1. Underline the payoff for player 1's best response.

2. Underline the payoff for player 2's best response.

Example - Payoff matrix with players Deepak and Amit choosing to contribute or not:

Example Explanation

• If Deepak contributes, Amit's best response is to contribute, yielding 4.

• If Deepak does not contribute, Amit also contributes to receive 2.

• To find Nash equilibrium, we look for cells where best response payoffs are underlined. Here, (Contribute, Contribute) is the Nash equilibrium.

12.3.3 No Nash Equilibrium

Consider another example where no Nash equilibrium exists:

Investigation reveals that behavior regarding strategy responses leads to no Nash equilibrium.

12.3.4 Dominant Strategy

Referring back to Deepak and Amit's example, (Contribute, Contribute) serves as both Nash equilibrium and dominant strategy. A dominant strategy is optimal regardless of the decisions taken by the other player.

Definition of Dominant Strategy: A choice by a player that is optimal regardless of competing decisions. Distinct from Nash equilibrium, which accounts for strategies based on the expected choices of others.

Insight Through Prisoners' Dilemma

The classic example illustrates two suspects choosing whether to confess or deny:

• If both confess, they serve 5 years.

• If one confesses while the other denies, the confessor gets 1 year, the denier gets 10.

• If neither confesses, they both get 2 years.

Payoff Matrix of the Dilemma:

Through reasoning, it can be shown confession is a dominant strategy for both parties despite leading to a Nash equilibrium of (Confess, Confess) resulting in greater jail time than an optimal alternative (No Confessions: 2 years).

12.3.5 Dominant Strategy of One Player Only

In certain games, only one player possesses a dominant strategy while the other player makes adjustments based on this:

Here, A (Player A) has a dominant strategy while B (Player B) adapts their responses, creating a Nash equilibrium based on A's choice.

12.3.6 Battle of Sexes

In this game, a husband and wife choose an activity to enjoy, either a movie or shopping, with different preferences:

None has a dominant strategy, yet Nash equilibria exist at mutual choices by couple.

In-Text Questions

1. Best responses for players?

2. Dominance dynamics in multiple strategies?

3. Impact of dominant strategies on Nash equilibrium?

12.4 Sequential Games

Sequential games involve players making decisions in a set order, where one player makes a move first and the other follows, possessing knowledge of the preceding actions.
Examples include:

• Chess, where players alternate moves based on knowledge of previous strategies.

• Market scenarios such as Ecolaza and Ecostic, where one firm reacts based on the decisions of competitors.

12.4.1 Solving Sequential Games

Sequential games can be illustrated through game trees, employing backward induction methods to forecast player strategies.
Example: Representing Ecolaza's market strategy against ThunderCo using a game tree, where decisions and respective payoffs are outlined.

12.4.2 Revisiting Battle of Sexes as Sequential Game

Considering the previous payoff matrix, the sequential nature alters player choices where one plays first. Solving through backward induction leads to predicted strategies and outcomes, resulting in a Nash equilibrium with defined payoffs.

In-Text Questions

5. Solving sequence game methods?

6. Impacts of sequence moves on strategies?

12.5 Summary

This chapter covered both simultaneous and sequential games, distinguishing between Nash equilibrium and dominant strategies. It highlighted the positions where a first mover may have advantages and outlined considerations for responding to competitors through informed choice.

12.1 Learning Objectives

• Analyze Simultaneous Games: Understand how players make decisions concurrently under imperfect information.

• Identify Nash Equilibrium: Determine the strategy profile where no player has an incentive to deviate unilaterally.

• Distinguish Dominant Strategies: Contrast dominant strategies (best regardless of others) with Nash Equilibrium (best given others).

• Examine Sequential Games: Explore games played in rounds where information about previous moves is available.

• Apply Backward Induction: Use reasoning from the end of a game to the beginning to solve for optimal sequential strategies.

12.2 Introduction

Game theory studies strategic interactions where the outcome for one player depends on the actions of others.

• Simultaneous Games: Players act at the same time or without knowing others' moves, resulting in imperfect information.

• Sequential Games: Players move in a specific order, creating perfect information (or at least knowledge of prior actions).

• Strategies:

  • Pure Strategy: A player chooses a single specific action (e.g., always choosing "Top").

  • Mixed Strategy: A player randomizes between actions based on probabilities (to be discussed later).

12.3 Simultaneous Games

In these games, players are effectively blind to the choice of their opponent at the moment of decision-making.

• Examples: Blind auctions, Rock-Paper-Scissors, or simultaneous price-setting by two companies.

• Discrete Strategy: A choice from a limited, finite set of actions (e.g., ${Heads, Tails}$).

12.3.1 The Payoff Matrix

A payoff matrix represents the possible outcomes. The first number in parentheses represents Player 1's (Row Player) payoff, and the second represents Player 2's (Column Player) payoff.

Analysis:

• If Player 1 chooses Bottom and Player 2 chooses Left, payoffs are $4$ for Player 1 and $2$ for Player 2.

• If Player 1 chooses Top and Player 2 chooses Right, payoffs are $3$ for Player 1 and $5$ for Player 2.

12.3.2 Nash Equilibrium

Definition: A Nash Equilibrium is a set of strategies where, given the strategies of all other players, no player can increase their payoff by changing only their own strategy. It is the "point of no regret."

Finding Nash Equilibrium (The Underline Method):

1. For Player 1 (Rows): For each column choice of Player 2, underline the highest payoff for Player 1.

   • If Player 2 picks Left: Player 1 compares $2$ vs $4$. Underline $4$ (Bottom).

   • If Player 2 picks Right: Player 1 compares $3$ vs $1$. Underline $3$ (Top).

2. For Player 2 (Columns): For each row choice of Player 1, underline the highest payoff for Player 2.

   • If Player 1 picks Top: Player 2 compares $3$ vs $5$. Underline $5$ (Right).

   • If Player 1 picks Bottom: Player 2 compares $2$ vs $1$. Underline $2$ (Left).

3. Result: A cell with both payoffs underlined is a Nash Equilibrium. In the matrix above, (Bottom, Left) is a Nash equilibrium ($4, 2$) because neither wants to move away given the other's choice.

12.3.3 Dominant Strategy

Definition: A strategy that earns a player a higher payoff than any other strategy, no matter what the other players do.

• Example (Deepak & Amit):

• If Amit contributes, Deepak prefers Contribute (4 > 1).

• If Amit doesn't contribute, Deepak still prefers Contribute (2 > 0).

• Result: "Contribute" is Deepak's dominant strategy.

12.3.4 Insight: The Prisoner's Dilemma

This paradox shows why two rational individuals might not cooperate even if it appears in their best interest.

• Matrix:

  • Both Confess: $(5, 5)$ years jail.

  • One Confesses, one Denies: $(1, 10)$ (Confessor gets $1$, Denier gets $10$).

  • Both Deny: $(2, 2)$ years jail.

• Logic: Even though $(2, 2)$ is better collectively, individual incentive leads both to "Confess" (Dominant Strategy), resulting in a $(5, 5)$ Nash Equilibrium. This highlights the tension between individual and collective rationality.

12.4 Sequential Games

In sequential games, players move at different times. The second mover has the advantage of observing the first mover's action.

• Representation: These are often shown using Game Trees (Extensive Form).

• Backward Induction: To solve, we look at the last decision in the tree and determine what that player would do. Then, move backward to the previous player's decision, knowing what the last player will eventually choose.

12.4.1 Battle of the Sexes (Sequential Version)

• Context: A couple prefers different activities (Movie vs. Shopping) but prefers being together over being apart.

• Sequential Logic: If the Husband chooses "Movie" first and the Wife sees this, she may choose "Movie" as well because $(Movie, Movie)$ is better for her than being alone at "Shopping," even if she prefers Shopping. This is known as the First-Mover Advantage.

12.5 Summary

• Simultaneous Games are solved using payoff matrices to find Nash Equilibria.

• Sequential Games are solved using backward induction on game trees.

• Nash Equilibrium is a state of mutual best responses.

• Dominant Strategy is a move that is always the best, regardless of the opponent's actions.