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In: Economics

Two people are fighting over an object and they each have a dollar which can be...

Two people are fighting over an object and they each have a dollar which can be allocated towards the fight either entirely or partially. They each face the following decision: How much of the dollar to allocate towards the fight and how much of the dollar they should use productively. Person 1 can denote amount x (out of the dollar he has) towards the fight and person 2 can denote the amount y (out of the dollar) towards the fight. Production is determined by the following equation: Output produced= 2-x-y Production is shared according to the following rules: 1. The player who will allocate a larger proportion of the dollar towards the fight will obtain the entire produce and the other player does not get anything. 2. If they allocate equal amounts, the entire produce is split equally amongst them Each player wants to maximize their share of output. Model this as a strategic game and solve for all pure strategy Nash equilibria. What are the payoffs to player 1 and 2 in equilibrium

Solutions

Expert Solution

According to the given situation, it is given that:

  • Person 1 will denote the x amount towards the fight, and
  • Person 2 will denote the y amount towards the fight.

Therefore the strategic game can be modelled as:

Here, first identify the optimal strategy of both the person and then determine the pure strategy Nash equilibrium.

1. If person 1 expects person 2 to fight, person one will also choose to fight as 2 is greater than 0. On the ohther hand, if person 1 expects person 2 to produce, person one will choose to fight as it gives higher payoff than choosing production ( 4 is greater than 3).

Similar analysis is used for person 2 such as:

If person 2 expects person 1 to fight, person one will also choose to fight as 2 is greater than 0. While, if person 2 expects person 1 to produce, person one will choose to fight as it gives higher payoff than choosing production ( 4 is greater than 3).

2. According to the analysis, there is only a single pure strategy Nash equilibrium that is (fight, fight) as this is the best payoff for both the person and both will not choose other payoff straegy over this strategy. Hence, the payoffs to player 1 and 2 in equilibrium are 2,2.

Rahul Sunny answered 1 year ago
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