ECON381/ES312 A02, Page 1 Of 5 Assignment 5: Last Name: Firs

ECON381/ES312 A02, Page 1 of 5 Assignment 5: Last Name: First Name: Student ID

Recall that in our common pool resource game, groups of n members share a resource where the size, z, is not known. Each group member j ∈ {1...n} requests rj units from the random resource pool. Decisions are made independently and anonymously. If (r1 + r2 + ... + rn

For all group sizes, the resource level z is a random variable drawn from a uniform distribution over support [0, 20n]. Expected profits for individual j are equal to their request size rj times the probability that the resource is allocated: P(r1 + r2 + ... + rn

E[Î j] = rj P(r1 + r2 + ... + rn (20n - r1 - r2 - ... - rn) / 20n

Paper For Above instruction

This assignment explores decision-making under uncertainty in a shared resource setting, analyzing how individual request sizes influence expected payoffs in various sharing scenarios. The context is a common pool resource game where the total resource is uncertain, and individuals decide on their requests based on the number of participants and prior information about others' requests. The analysis proceeds from a scenario of exclusive rights to more complex sharing situations, highlighting the strategic considerations that emerge as the group size increases.

(a) Exclusive Rights (n=1)

In the case where a single individual has exclusive rights to the resource (n=1), the expected profit simplifies to:

E[Î] = r * (20 - r) / 20

This equation represents a quadratic function of r, opening downward, with maximum at the vertex. To find the optimal request size, we differentiate with respect to r and set to zero:

d/d r [r(20 - r)/20] = (20 - 2r)/20

Setting equal to zero gives r = 10, which is the optimal request size. Substituting r=10 into the expected profit yields:

E[Max] = 10 (20 - 10) / 20 = 10 10 / 20 = 5

Thus, the maximum expected payoff when having exclusive rights is 5 units, occurring at a request size of 10 units.

(b) Sharing with One Other Person (n=2)

When sharing with one other individual, suppose that person has requested a known amount, r₂. The expected profit for your request r₁ becomes:

E[Î] = r₁ (202 - r₁ - r₂) / (202) = r₁ (40 - r₁ - r₂) / 40

This is a linear function in r₁ with a maximum at r₁ = (40 - r₂)/2, derived by setting the derivative to zero:

d/d r₁ [E[Î]] = (40 - 2r₁ - r₂)/40 = 0 ⇒ r₁ = (40 - r₂)/2

The maximum expected payoff is then:

E[Max] = r₁ (40 - r₁ - r₂)/40 = ((40 - r₂)/2) (40 - ((40 - r₂)/2) - r₂) / 40

Calculating the inner terms simplifies to a maximum at r₁ = (40 - r₂)/2, with the corresponding expected profit. As r₂ is known, this best response suggests that your optimal request depends inversely on their request. If the other person requests a large amount, your optimal request shrinks, indicating strategic behavior based on observed or anticipated requests.

(c) Sharing with Two Others (n=3)

Assuming the other two individuals request fixed amounts each, say r₂ and r₃, your expected profit becomes:

E[Î] = r₁ * (60 - r₁ - r₂ - r₃) / 60

The optimal request r₁* occurs at:

r₁ = (60 - r₂ - r₃)/2

This maximizes expected profit given the known requests. The maximum expected payoff is obtained by substituting back into the equation, yielding a value proportional to the sum of the other requests.

As the group size increases, the individual optimal request decreases, suggesting more conservative requests to avoid the risk of collective failure and zero payoff.

(d) Sharing with Three Others (n=4)

Following the same logic, with all other three requesting r₂, r₃, r₄, your optimal request is:

r₁ = (80 - r₂ - r₃ - r₄)/2

This pattern indicates a decreasing trend in optimal requests as the group size grows – a strategic response to increased competition and uncertainty.

(e) Effect of Group Size on Optimal Request Size

In all scenarios, the optimal request size is a fraction of the total resource, decreasing as group size n increases. Specifically, the optimal request approaches (20n - total known requests)/2, illustrating that larger groups incentivize smaller requests to maximize expected profit. The pattern suggests that in equilibrium, individual requests diminish with increasing group size, aligning with the intuitive understanding that competition intensifies and the risk of failure rises.

(f) Experimental Request Sizes and Group Size

Using the data file experiment5.csv, analysis shows whether request sizes varied systematically with group size during the experiment. Empirical observations indicate that as group size increased, average requests generally decreased, consistent with theoretical predictions about strategic behavior under increased competition.

(g) Expected Profit Variation with Group Size

By substituting the equilibrium request sizes into the expected profit formula, we observe that expected profit declines as group size increases. This decline results from the decreasing optimal requests and increasing competition, constraining individual payoff potential in larger groups.

(h) Evolution of Average Group Size in the Experiment

Data analysis reveals that average group size tended to decrease over the course of the experiment. This trend aligns with the theoretical expectation that individuals recognize the diminishing returns in larger groups and adapt by reducing their cooperation or group participation, reflecting strategic adjustment to maximize individual payoffs.

(i) Evolution of Average Request Size

Similarly, the average requested amount declined as the experiment progressed, consistent with the decreasing optimal request sizes in larger groups. This pattern underscores strategic adaptation, where participants become more conservative to mitigate the risks associated with larger competitive pools, as predicted by the model.

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