Game Theory Analysis: Sequential Deviations

The scenario has changed dramatically. Instead of a single deviation in the final round, we now have a pattern: four different actors deviate in sequence during the odd-numbered scenes (1, 3, 5, and presumably 7, though only 6 scenes exist). This structure suggests a coordinated or predictable pattern of behavior, which fundamentally alters the game-theoretic analysis.

The New Strategic Landscape

The original model of a one-off final deviation is no longer applicable. The players are not reacting myopically to a final stage; they are participating in a dynamic process with a clear pattern of action.

Primary Game Type: Dynamic Game with a State Variable

This is best analyzed as a dynamic game or a sequential game with a state. The "state" of the game is the history of who has deviated and when. Each player's strategy can now be conditioned on this state.

The fact that deviations are alternating and by different players implies this is not random defection but part of a larger, predefined strategy profile or mechanism.

Possible Interpretations and Analyses

The nature of the analysis now depends on why the deviations are happening in this specific sequence.

Interpretation 1: A Pre-Arranged Mechanism (Cooperative Strategy)

This is the most likely analysis if the outcome is still successful. The "script" is not just the lines but the entire plan, which includes these scheduled deviations.

• Strategy: The group has collectively agreed to a plan where specific actors deviate in specific scenes to achieve a greater overall payoff (e.g., to create a specific artistic effect, build tension, or surprise the audience).
• Game Theory Model: This can be seen as a correlated equilibrium on a grand scale. The players are using an external signal (the scene number and their assigned role) to coordinate their actions. No one is "deviating" from the true plan; they are following a more complex, pre-arranged set of strategies.
• Result: The notation would show this as the intended outcome. The "deviations" are part of the equilibrium strategy, not a break from it.

Interpretation 2: A Tit-for-Tat Punishment Chain (Non-Cooperative)

This interpretation applies if the deviations are punitive responses to prior actions within the game.

• Scenario: Perhaps Actor A deviated in an unseen "pre-scene" or there's an initial provocation. Actor 1 deviates in Scene 1 as punishment. This punishment then must itself be punished by Actor 2 in Scene 3, and so on, creating a chain reaction of retaliation.
• Game Theory Model: This resembles a trigger strategy or a punishment cycle in a repeated game. The game becomes a series of reciprocal actions.
• Result: The notation would show a cascade of defections leading to a poor overall outcome for all involved, a classic result of ongoing retaliation in games like the Prisoner's Dilemma.

Interpretation 3: A Signalling or Screening Game

The deviations could be a way for actors to signal private information (e.g., their skill, commitment, or dissatisfaction) to the director, audience, or other actors.

• Model: This would be analyzed using the tools of signalling games. An actor chooses to deviate (a costly action) to reveal their type or to force a reaction.
• Example: A lesser-known actor deviates in Scene 1 in a flashy way to "signal" their talent and grab attention. The next deviator might be signaling something else.

Comparison of the Two Scenarios

Feature	Original Scenario (One Final Deviation)	New Scenario (Sequential Deviations)
Game Type	Finitely Repeated Game	Dynamic Game / Mechanism
Solution Concept	Backward Induction, SPNE	Correlated Equilibrium, Signalling Equilibrium, Trigger Strategies
Nature of "Deviation"	A true break from the strategy, leading to a new Nash Eq.	Likely part of a larger, pre-arranged coordinated strategy.
Information	Complete Information	May involve Incomplete Information (Signalling)
Predicted Outcome	Suboptimal due to last-minute defection	Potentially Optimal, if the mechanism is well-designed

Conclusion

Yes, this fundamentally changes the analysis. A single late deviation is a classic result of non-cooperation in a finite game. In contrast, a scheduled, alternating sequence of deviations by different actors indicates a far more complex and likely cooperative strategy is at play.

The game is no longer a simple repeated structure but a dynamic game best analyzed as a pre-arranged mechanism (correlated equilibrium) or a signalling game. The "deviations" are not necessarily deviations from the true plan; they are the plan. The notation would therefore represent the successful execution of a sophisticated coordinated strategy, not a breakdown of cooperation.