Scene At The End Of The Princess Bride: Wesley The Hero

In A Scene At The End Of The Princess Bridethe Hero Wesley Confron

In a scene at the end of The Princess Bride, the hero, Wesley, confronts the evil Prince Humperdinck. This interaction can be modeled as a game where Wesley can either be strong or weak, with the probability of each assigned equally. Wesley, knowing his own type, chooses to stand or lie in bed. The prince observes Wesley's decision but does not know his type and then chooses to fight or surrender. Payoffs depend on Wesley's type, his action, and the prince’s response, with additional considerations such as the cost for weak Wesley to stand. This paper will formalize the game in extensive form, derive a pooling Bayesian Nash equilibrium with initial belief q=0.5, and identify conditions for a separating Bayesian Nash equilibrium.

Paper For Above instruction

Extensive Form of the Game

The extensive form of the game begins with nature's move where Wesley's type is determined: with probability 0.5, Wesley is weak, and with probability 0.5, he is strong. This initial move is not observed by the prince. Wesley then observes his own type and chooses an action: to stand or to lie in bed.

Following Wesley's action and his known type, the prince observes only the action and not Wesley’s type. The prince then chooses to fight or surrender. The payoffs depend on Wesley's type, his action, and the prince’s decision, encapsulating the strategic interaction. For clarity, the extensive form can be depicted as:

1. Nature chooses Wesley's type (Weak or Strong) with equal probability (0.5 each).

2. Wesley observes his type and chooses an action: Stand (S) or Lie in bed (B).

3. The prince observes Wesley's action but not his type.

4. The prince chooses to Fight (F) or Surrender (R).

The payoffs are as follows:

- If Wesley is weak:

- Standing and fighting: Payoff to Wesley = -c + 3 (benefit from beating the prince, minus the cost c), payoff to prince = -2.

- Standing and surrender: Payoff to Wesley = 1, payoff to prince = 0.

- Lying in bed and fighting: Payoff to Wesley = 1, payoff to prince = -1.

- Lying in bed and surrender: Payoff to Wesley = 1, payoff to prince = 0.

- If Wesley is strong:

- Standing and fighting: Payoff to Wesley = 3, payoff to prince = -2.

- Standing and surrender: Payoff to Wesley = 1, payoff to prince = 0.

- Lying in bed and fighting: Payoff to Wesley = 1, payoff to prince = -1.

- Lying in bed and surrender: Payoff to Wesley = 1, payoff to prince = 0.

This extensive form captures all the strategic choices and their consequences, encompassing the variability introduced by Wesley's type and the imperfect information available to the prince.

Pooling Bayesian Nash Equilibrium (q=0.5)

A pooling Bayesian Nash equilibrium involves Wesley taking the same action regardless of his type, convincing the prince that Wesley's action does not reveal his type. With the prior belief q=0.5, the prince considers the expected payoffs from fighting or surrendering based on observing Wesley's action.

Prince’s beliefs based on observation:

- If Wesley is observed lying in bed, the probability that Wesley is weak is the prior, 0.5.

- If Wesley is observed standing, the equally prior belief remains, 0.5 that Wesley is weak, unless Wesley’s strategy minimizes type differentiation.

Equilibrium Conditions:

- Wesley's choice: To pool actions, he chooses a single action (either always stand or always lie in bed). To make the equilibrium valid, Wesley's utility from this action must be at least as high as deviating to the other action, given the expected payoff considering the prince's responses.

- Prince’s decision: Given observation, the prince's expected payoff from fighting is calculated based on the posterior belief about Wesley's type.

Suppose Wesley always lies in bed (B). The expected payoffs are:

- For Wesley (weak or strong): Payoff from lying in bed is 1 or 1, respectively.

- The prince believes Wesley is equally likely to be weak or strong, each with probability 0.5, so the expected payoff for fighting is:

\[

0.5 \times (-1) + 0.5 \times (-2) = -1.5

\]

- The payoff for surrender is always 0 (prince) or 1 (Wesley), which is better for Wesley if he expects the prince to surrender.

In equilibrium, the prince prefers surrender over fight if the expected payoff from fighting is less than or equal to 0.

Similarly, if Wesley always stands (S), the equations for expected payoffs are:

- For Wesley: payoffs are 1 or 3.

- The prince’s expected payoff from fighting is:

\[

0.5 \times (-2) + 0.5 \times (-2) = -2

\]

- If the prince expects Wesley to always stand, surrender yields 0 (prince) or 2/1 (Wesley), which influences his decision.

Range of c:

The value of c affects Wesley’s incentives to stand vs. lie in bed when weak. If c is too large, Wesley prefers lying in bed regardless of his type, maintaining a pooling equilibrium at bed. For the belief to be valid (Wesley indifferent or prefers pooling), the cost c must satisfy:

\[

c \leq 2

\]

since standing costs the weak Wesley an additional c, and he chooses to pool in bed when the cost outweighs the benefits from standing.

Conclusion:

The pooling equilibrium exists when Wesley's actions do not reveal his type, which is sustained if the cost c is sufficiently high (c ≥ 2) to prevent weak Wesley from standing, thus maintaining the pooling in bed strategy.

Separating Bayesian Nash Equilibrium and c-range

A separating Bayesian Nash equilibrium occurs when Wesley's actions perfectly reveal his type: weak or strong. For this, Wesley chooses to stand only if strong, and stays in bed only if weak, allowing the prince to infer Wesley's type accurately.

Wesley's strategies:

- Weak Wesley: stay in bed.

- Strong Wesley: stand.

Prince’s response:

- If observing Wesley in bed: Belief that Wesley is weak is 1, so the prince surrenders.

- If observing Wesley standing: Belief that Wesley is strong is 1, so the prince fights.

Incentive compatibility:

- Weak Wesley prefers to stay in bed if the payoff from lying in bed exceeds the payoff from standing, considering the cost c:

\[

\text{Lying in bed payoff} = 1 \quad,\quad \text{Standing payoff} = -c + 3

\]

Weak Wesley prefers lying in bed when:

\[

1 \geq -c + 3 \Rightarrow c \geq 2

\]

- Strong Wesley prefers to stand as it yields 3, which is better than lying in bed (which yields 1).

Range of c:

- To sustain the separating equilibrium, the cost c for weak Wesley must be sufficiently high (c ≥ 2), making standing unattractive for the weak type and incentivizing perfect separation. For c

Conclusion:

The game exhibits multiple equilibria depending on the value of c: a pooling equilibrium when c is high (c ≥ 2) in which Wesley's action conceals his type, and a separating equilibrium when c is low (c

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