Game Theory Discussion from The Liar's Bar

Game Theory is both a new branch of modern mathematics and an important discipline in operations research.

Game Theory primarily includes the following elements:

1. Players: In a competition or game, each participant with decision-making power is called a player. Games with only two players are called "two-person games," while those with more than two players are called "multiplayer games."

2. Strategy: In a game, each player has a complete set of feasible action plans. A strategy is not just a plan for a specific stage but a comprehensive plan guiding the entire action. If players have a finite number of strategies, it's called a "finite game"; otherwise, it's an "infinite game."

3. Payoff: The outcome at the end of a game is called the payoff. Each player's payoff depends not only on their own chosen strategy but also on the strategies chosen by all other players. Therefore, each player's "payoff" is a function of the set of strategies chosen by all players.

4. Outcome: For game participants, there exists a game outcome.

In The Liar's Bar, the players are the participants, strategies involve choosing to play cards or challenge based on previous plays and other players' actions, and payoffs/outcomes determine whether one forces others to take the gun or takes it themselves.

Some interesting concepts in Game Theory:

1. Nash Equilibrium Nash Equilibrium refers to a situation where all participants face a scenario where their current strategy is optimal given others' strategies. At Nash Equilibrium, no rational participant would unilaterally change their strategy.

The famous "Prisoner's Dilemma" exemplifies this concept. Two thieves are interrogated separately. If both confess, each gets 8 years; if one confesses while the other denies, the confessor goes free while the denier gets 10 years; if both deny, each gets 1 year.

In this dilemma, "mutual betrayal" is a Nash Equilibrium. When A betrays, B's best strategy is to betray; when B betrays, A's best strategy is also to betray. Though this outcome is worst for them collectively, individual rationality drives them to this equilibrium.

2. Zero-Sum Games: In zero-sum games, under strict competition, one player's gain exactly equals another's loss, with the total sum always being "zero." There's no possibility of cooperation for mutual benefit.

Clearly, games in The Liar's Bar are zero-sum games - there must be winners and losers, with no possibility of mutual victory.

Let's analyze The Liar's Bar's poker mode:

Strategy Space:

• Honest Play: Playing cards and declaring true values (A, K, Q). Advantages include smooth gameplay without life risk; disadvantages include potentially missing play opportunities.
• Deceptive Play: Playing cards while declaring false values. This strategy might gain advantages but risks Russian roulette if caught.

Response Strategies:

• Challenge Strategy: Players can challenge others' declarations. Successful challenges force liars into Russian roulette; failed challenges may damage trust.
• Non-Challenge Strategy: Accepting others' declarations maintains smooth gameplay but might allow deception to succeed.

Payoff Analysis: Honest Play Payoffs:

• With honest opponents: Steady gameplay with gradual advantage building
• Against successful liars: Potential disadvantage in current situation

Deceptive Play Payoffs:

• If successful: Quick tactical advantages
• If caught: Risk of Russian roulette, potentially game-ending consequences

Challenge Payoffs:

• Direct Benefits: Successful challenges may eliminate competitors or deplete their safe shots
• Reputation Benefits: Builds image as skilled player
• Game Control Benefits: Ability to influence game pace and direction

Challenge Risks:

• Direct Risk: Russian roulette if challenge fails
• Trust Damage: Failed challenges harm credibility
• Strategy Exposure: May reveal strategic tendencies

Nash Equilibrium Analysis:

1. Pure Strategy Nash Equilibrium

• All-Honest Strategy: Can form equilibrium as deviation risks Russian roulette
• All-Deceptive Strategy (Theoretical): Possible but unstable in practice

2. Mixed Strategy Nash Equilibrium Assuming two players with probabilities p and q for honest play: E1 = pq × Rhh + p(1-q) × Rhl + (1-p)q × Rlh + (1-p)(1-q) × Rll Where R represents various payoff combinations.

3. Bayesian Considerations Players update beliefs about opponents' honesty using Bayesian inference based on:

• Prior probability of deception
• Card distribution knowledge
• Behavioral cues
• Declaration patterns

For example, if many Aces have been played, a new Ace declaration might increase estimated deception probability, influencing challenge decisions through Bayesian expected payoff calculations.