Question

Suppose that two people (person 1 and person 2 ) are considering whether to form a partnership firm. Person 2 's productivity (type) is unknown to person 1 at the time at which these people must decide whether to create a firm, but person 1 knows that, with probability \(p\), person 2 's productivity is high (H) and, with probability \(1-p\), person 2's productivity is low (L). Person 2 knows her own productivity. If these two people do not form a firm, then they each receive a payoff of 0 . If they form a firm, then their payoffs are as follows: If person 2's type is \(\mathrm{H}\), then each person receives 10. If person 2's type is \(L\), then person 2 receives 5 and person 1 receives \(-4\) (which has to do with person 1 having to work very hard in the future to salvage the firm). (a) Consider the game in which person 1 chooses between forming the firm (F) or not forming the firm (O). Draw the extensive form of this game (using a move of nature at the beginning to select person 2's type). Note that only one person has a move in this game. What is the Bayesian Nash equilibrium of the game? (It depends on \(p\).) (b) Now suppose that before person 1 decides whether to form the firm, person 2 chooses whether or not to give person 1 a gift (such as a dinner). Player 1 observes person 2's choice between \(\mathrm{G}\) and \(\mathrm{N}\) (gift or no gift) before selecting F or O. The gift entails a cost of \(g\) units of utility for person \(2 ;\) this cost is subtracted from person 2 's payoff designated earlier. The gift, if given, adds \(w\) to person 1's utility. If person 2 does not give the gift, then it costs her nothing. Assume that \(w\) and \(g\) are positive numbers. Draw the extensive form of this new game. (c) Under what conditions (values of \(g\) and \(w\) ) does a (separating) perfect Bayesian equilibrium exist in which the low type of person 2 does not give the gift, the high type gives the gift, and person 1 forms the firm if and only if a gift is given? Completely specify such an equilibrium. (d) Is there a pooling equilibrium in this game? Fully describe it and note how it depends on \(p\).

Short answer

(a) Equilibrium: Form firm if \(p \geq 2/7\). (c) Separating: Exists if \(g < 5\), \(w > 4\). (d) Pooling: Possible if gift given always, dependent on \(p\).

Step-by-step solution

Identify Possible Outcomes

There are two possible outcomes for the game based on person 2's productivity, which is either high (H) with probability \( p \) or low (L) with probability \(1-p \). If person 2's type is H and the firm is formed, the payoff for both is 10. If her type is L and the firm is formed, person 1's payoff is -4 and person 2's payoff is 5.

Establish Person 1's Strategy

Person 1 decides to form the firm (F) or not form the firm (O). The decision depends on expectations: Form if the expected payoff when forming the firm is greater than zero.

Calculate Expected Payoffs

The expected payoff for forming the firm is \( 10p - 4(1-p) = 14p - 4 \). The payoff for not forming the firm is 0. Person 1 forms the firm if \( 14p - 4 \geq 0 \), simplifying to \( p \geq \frac{2}{7} \).

Understand the Bayesian Nash Equilibrium

The Bayesian Nash Equilibrium occurs when person 1 forms the firm if \( p \geq \frac{2}{7} \) and does not form it if \( p < \frac{2}{7} \).

Analyze Game with Gift (G) Option

In the new game, person 2 chooses to give a gift (G) or not (N) before person 1 decides F or O. This introduces additional utility variables: \( w \) for person 1 if gift given and \( g \) cost for person 2.

Determine Conditions for Separating Equilibrium

There is a separating equilibrium where the high type gives the gift, and the low type doesn't if forming the firm after a gift is beneficial for person 1. Solve the system of inequalities by setting \(10 - g > 10w\) for the high type and \(-4 - g < -4w\) for the low type. If \(g < 5\) and \(w > 4\), the conditions hold.

Consider Pooling Equilibrium

In a pooling equilibrium, person 2 gives a gift regardless of type, with person 1 forming the firm based on perceived probabilities. This equilibrium exists and depends on \(p\) such that the expected payoff of forming the firm outweighs the cost of gift by the population predicted with probability \( p \).

Key concepts

Bayesian Nash equilibrium

The Bayesian Nash equilibrium is a fundamental concept in game theory which extends the notion of Nash equilibrium to games involving incomplete information. In our scenario, the uncertainty involves person 1 not knowing person 2's productivity level, which can be either high or low. Bayes' theorem is used to update beliefs about these uncertain events.

The Bayesian Nash equilibrium in this context is the strategy profile for person 1 that maximizes their expected payoff given their beliefs about person 2's type (high or low productivity) and person 2's type-dependent strategies. In this solution, the equilibrium implies that person 1 chooses to form the firm (F) if the probability of person 2 being highly productive (H) is sufficiently high. Specifically, person 1 forms the firm if the probability, denoted by \( p \), of person 2 being high, is greater than or equal to \( \frac{2}{7} \). This condition ensures that expected benefits outweigh the risks of person 2 being of low productivity, where the payoffs differ significantly.

• Person 1 assumes \( p \) probability for high productivity and \( 1-p \) for low productivity
• An equilibrium strategy depends on formation benefits outweighing risks
• Bayesian Nash equilibrium needs balancing both probabilistic beliefs and strategic responses

extensive form game

The extensive form game is a way to represent games visually using a tree-like graph to show decision points and possible outcomes. It effectively captures the sequence of actions, the information available at each decision node, and the resulting payoffs.

In this problem, the extensive form diagram starts with a "nature" move indicating the probability \( p \) that person 2 is a high type (H) and \( 1-p \) that they are low (L). From here, person 2 may give or not give a gift, after which person 1 makes their decision to form (F) or not form a firm (O). The diagram shows these decisions branching out with potential payoffs listed at each 'leaf' or terminal node of the tree notation. These diagrams help break down complex strategic interactions into manageable parts.

• Initial decisions influenced by chance (probability \( p \))
• Branches represent strategic decisions by players
• Terminal nodes indicate resultant payoffs for each outcome

perfect Bayesian equilibrium

The perfect Bayesian equilibrium is a refined concept of equilibrium in games with incomplete information. It combines the ideas of Nash equilibrium with Bayes-rational beliefs about uncertainties. This equilibrium concept allows players to update their beliefs based on the observed actions of others according to Bayes' rule.

To find a perfect Bayesian equilibrium, we consider how the introduction of gifts impacts strategic decisions in our game. The separating equilibrium described requires that the low type does not give a gift, the high type does, and person 1 forms a firm if and only if a gift is given. This setup implies that person 1 can update beliefs based on the gift action, matching probabilities to observed outcomes.

• High productivity types provide gifts, signaling their type
• Equilibrium establishes belief consistency with observed strategy
• Refines belief updating about player types through strategic choices

pooling equilibrium

Pooling equilibrium occurs when all types of a player choose the same action, making it impossible for the other player to distinguish between types based on their actions.

In our scenario, a pooling equilibrium can occur when both high and low types of person 2 choose to give a gift, leaving person 1 unable to differentiate based on this signal. The decision of person 1 then revolves around their belief about the probability \( p \) and the expected payoff.

If the population's predicted probability of high productivity, coupled with the average utility benefit of gift giving, results in forming the firm being more rewarding than the no-formation payoff, pooling equilibrium can exist. Here, person 1 bases decisions on the average expected utility, disregarding individual signals from types because they don't offer additional information.

• Every type of player sends the same signal
• Person 1 cannot tell types apart based on gifts
• Key determinant: expected average payoff of strategies