So the monarchy vs tit for that that allows for mistakes that emerge into a socialist structure tradeoff is measured by this: (I love that term 'cost of anarchy' BTW)

The "cost of anarchy" in mechanism design refers to the inefficiency that arises in a system when individuals act in their own self-interest rather than cooperating for a collective good. It is often quantified in terms of the difference between the social welfare achieved under a centralized, optimal mechanism and the social welfare achieved under a decentralized mechanism where agents act independently.

In more formal terms, the cost of anarchy is typically defined as the ratio of the worst-case social welfare (or utility) achieved by a Nash equilibrium of a game to the optimal social welfare that could be achieved with a centrally designed mechanism. This concept is particularly relevant in the context of game theory and economics, where it highlights the potential inefficiencies that can arise in competitive environments.

Mathematically, if \( SW^* \) is the optimal social welfare and \( SW_{NE} \) is the social welfare at a Nash equilibrium, the cost of anarchy (CoA) can be expressed as:

\[

\text{Cost of Anarchy} = \frac{SW_{NE}}{SW^*}

\]

A CoA greater than 1 indicates that the decentralized outcome is less efficient than the optimal outcome, emphasizing the potential losses due to individualistic behavior in the absence of coordination or regulation. Understanding the cost of anarchy helps in designing mechanisms that can mitigate these inefficiencies, such as through incentives or regulations that align individual interests with social welfare.