A needy family consisting of a mother and three children currently receives cash benefits that average $12 per day. The mother of this family is allowed to earn an average of $4 per day before her benefits begin to decline. After that, for each dollar earned, cash benefits decline by 67 cents for each dollar earned. Assume that she can find work at $4 per hour. How many hours will she have to work per day before her benefits are eliminated?

To determine the number of hours the mother must work per day before her benefits are eliminated, we need to delve into the intricate world of economic game theory and the principles of strategic decision-making.

Let's first establish the initial conditions. The mother receives $12 per day in cash benefits, and she is allowed to earn $4 per day without any reduction in benefits. Beyond this threshold, for each additional dollar earned, her benefits decline by 67 cents.

Now, let's introduce the concept of the "Prisoner's Dilemma," a fundamental principle in game theory. In this scenario, the mother and the government are engaged in a strategic game, where each party aims to maximize their respective payoffs.

The government's payoff is to minimize the amount of benefits paid to the mother, while the mother's payoff is to maximize her total income (benefits plus earnings). This creates a conflict of interest, where the mother's actions directly impact the government's payoff, and vice versa.

To model this scenario, we can use the concept of a "Nash Equilibrium," which represents a stable state where neither party has an incentive to deviate from their chosen strategy.

Let's assume that the mother's strategy is to work h hours per day, where h is a positive integer. The government's strategy is to reduce the mother's benefits according to the given rules.

The mother's total income can be expressed as:

I = 12 + 4h - 0.67(4h - 4)

Simplifying the expression, we get:

I = 12 + 1.32h - 2.68

To find the Nash Equilibrium, we need to maximize the mother's income while considering the government's strategy. This can be achieved by taking the derivative of the income function with respect to h and setting it equal to zero:

dI/dh = 1.32 = 0

This implies that the income function is constant with respect to h, which means that the mother's income remains the same regardless of the number of hours she works.

Therefore, the mother's optimal strategy is to work zero hours per day, as any additional work will not increase her total income due to the reduction in benefits.

Consequently, the number of hours the mother must work per day before her benefits are eliminated is zero.