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How can you evaluate the performance of an algorithmic game theory mechanism?

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Algorithmic game theory is a branch of operations research that studies how to design and analyze algorithms for strategic interactions among rational agents. A mechanism is a rule that specifies how the agents' actions and preferences are mapped to outcomes and payments. Evaluating the performance of a mechanism is crucial for understanding its efficiency, fairness, and incentives.

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Charles Igah

PhD Candidate at Universidad de Navarra| Researcher at CEIT
Chawanangwa Gondwe

Clinic Nurse - Kamuzu Central Hospital || Lighthouse Trust || MBA-Hospital Administration|| Epidemiology ||Biostatistics

1 What are the criteria for evaluating a mechanism?

One way to evaluate a mechanism is to compare it with an ideal benchmark, such as the social welfare, which is the sum of the agents' utilities. A mechanism is said to be optimal if it maximizes the social welfare. However, achieving optimality may require unrealistic assumptions, such as perfect information, unlimited computation, and truthful reporting. Therefore, other criteria are often used to measure how well a mechanism approximates the optimal outcome, such as the approximation ratio, the price of anarchy, and the price of stability.

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Charles Igah

PhD Candidate at Universidad de Navarra| Researcher at CEIT
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It’s important to to note that evaluating the performance of a mechanism in algorithmic game theory can be quite complex. In addition to the basic criteria like efficiency, individual rationality, budget balance, and other criteria like dynamic consistency, regret maximization, PoS, PoA. Here are my suggestions: 1. Learning Efficiency: this is used to measure how quickly and effectively a mechanism can adapt or learn in an environment with changing conditions or incomplete information. 2. Resilience to Collusion: Here we test the mechanism in scenarios where collusion is possible, and observe if the mechanism can still ensure fairness and efficiency. Others: Robustness to information asymmetry, Adaptability to heterogeneous preferences.

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Chawanangwa Gondwe

Clinic Nurse - Kamuzu Central Hospital || Lighthouse Trust || MBA-Hospital Administration|| Epidemiology ||Biostatistics
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The price of anarchy measures the ratio of the worst-case social welfare under the mechanism to the optimal social welfare that a centralized authority can achieve. A lower price of anarchy signifies a better mechanism. Conversely, price of stability measures the ratio of the best-case social welfare under the mechanism to the optimal social welfare, and a higher price of stability indicates a better mechanism. . Additionally, one can evaluate the mechanism's performance using simulation, wherein the mechanism is simulated with different inputs, and the resulting outputs are observed. This can help identify the mechanism's strengths and weaknesses and suggest improvements.

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2 What is the approximation ratio?

The approximation ratio of a mechanism is the ratio between the social welfare of the mechanism and the social welfare of the optimal mechanism. For example, if a mechanism achieves a social welfare of 80 and the optimal mechanism achieves a social welfare of 100, then the approximation ratio is 0.8. The closer the approximation ratio is to 1, the better the mechanism performs. The approximation ratio can be computed for different scenarios, such as the worst-case, the average-case, or the best-case.

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Chawanangwa Gondwe

Clinic Nurse - Kamuzu Central Hospital || Lighthouse Trust || MBA-Hospital Administration|| Epidemiology ||Biostatistics
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The approximation ratio is a measure of how close an approximate solution is to the optimal solution for a given optimization problem. It is expressed as a ratio or factor that guarantees the returned solution is within a certain distance of the optimal one. The guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor. For example, if an algorithm has an approximation ratio of `c - ?` for a maximization problem, it means that the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of `c - ?` of the returned solution.

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3 What is the price of anarchy?

The price of anarchy of a mechanism is the ratio between the social welfare of the optimal mechanism and the social welfare of the worst possible equilibrium of the mechanism. An equilibrium is a state where no agent has an incentive to deviate from its action. For example, if the optimal mechanism achieves a social welfare of 100 and the worst equilibrium of the mechanism achieves a social welfare of 50, then the price of anarchy is 2. The higher the price of anarchy, the more inefficient the mechanism is. The price of anarchy captures the loss of efficiency due to strategic behavior.

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4 What is the price of stability?

The price of stability of a mechanism is the ratio between the social welfare of the optimal mechanism and the social welfare of the best possible equilibrium of the mechanism. For example, if the optimal mechanism achieves a social welfare of 100 and the best equilibrium of the mechanism achieves a social welfare of 90, then the price of stability is 1.11. The lower the price of stability, the more efficient the mechanism is. The price of stability captures the gain of efficiency due to coordination.

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5 How can you compute these criteria?

To compute these criteria, you need to know the agents' utilities, the mechanism's outcome function, and the equilibrium concept. Depending on the complexity of the problem, you may need to use mathematical tools, such as linear programming, convex optimization, or fixed-point theory. Alternatively, you may use simulation methods, such as Monte Carlo sampling, agent-based modeling, or reinforcement learning. You may also need to consider different assumptions, such as complete or incomplete information, deterministic or stochastic outcomes, or single or multiple objectives.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How can you evaluate the performance of an algorithmic game theory mechanism?

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3

4

5

6

1 What are the criteria for evaluating a mechanism?

2 What is the approximation ratio?

3 What is the price of anarchy?

4 What is the price of stability?

5 How can you compute these criteria?

6 Here’s what else to consider

Operations Research

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How can you evaluate the performance of an algorithmic game theory mechanism?

1

2

3

4

5

6

1 What are the criteria for evaluating a mechanism?

2 What is the approximation ratio?

3 What is the price of anarchy?

4 What is the price of stability?

5 How can you compute these criteria?

6 Here’s what else to consider

Operations Research

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