# CGSS: Introduction to Mechanism Design

Game Theory outlines the problem of the free-rider dilemma in public goods. In order to overcome the tragedy of the commons mechanism design was proposed. The basic idea is to define the the payoff and actions in order to drive people towards a preferred behaviour.

Public Goods Game

From the mechanism design perspective  two individuals  $k \in \{i, j \}$ receive a private benefit $b_k$ at the cost of the public project $c \in \mathbb{R}_+$. A decision is denoted as $d\in \{ 0, 1 \}$ where 0 is the decision not to invest and 1 is the decision to invest. Each individual k pays a tax $t_k$. A decision is communicated via a message $M = M(m_i,m_j) = (d,t_i,t_j)$.

Based on this setup a good mechanism would at least consist of:

• Individual Rationality: individuals weakly prefer to participant
• Efficiency: maximizing the sum of utilities of all participating individuals
• Balanced: the taxes cover the costs
• Simplicity: easy to understand and practical (not well defined)

Additional properties could be defined, but are left for more in-depth studies.

Simple Public Goods Mechanism

Based on the work of Jackson & Moulin (1992) a two-stage mechanism can be defined. In the first stage each individual $k \in \{i, j \}$ submits a bit $v_k$ which is an estimate for the joint benefit of a project. If the largest bid $v^* > c$. In the second stage each individual $k \in \{i, j \}$ submits a bid $\beta_k$ indicating private benefit. If a cumulative private benefit $\sum_k \beta_k$ is larger than the cost $v^*$, the project should go ahead, if it is smaller, the project should not go ahead and equality leaves the decision undecided.

The mechanism has the properties of being individually rational, efficient, balanced, simplistic, implementable in dominant strategies, and offers true-preferences revelation.

Proof of optimality TBA.

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