the gears to ascension comments on Unifying Bargaining Notions (2/​2)

[the gears to ascension]

In both this and the previous post, this notation goes undefined:

$p^S\in \Delta {\prod }_{i\in S}{A}_i$

What are delta and product doing in this definition?

[the gears to ascension]

Via Phind.com in gpt4 mode:

In the given expression ${\rho }^S\in \Delta {\prod }_{i\in S}{A}_i$, the delta symbol ($\Delta$) represents the simplex, and the product symbol ($\prod$) denotes the Cartesian product.

The simplex, denoted by $\Delta$, is a concept from geometry, but in this context, it’s used to indicate the set of probability distributions over a finite set. In the context of game theory, the simplex is used to represent mixed strategies, which are probability distributions over the set of pure strategies.

The Cartesian product, denoted by $\prod$, is a mathematical operation that returns a set from multiple sets, where the elements of the resulting set are all possible ordered pairs (or tuples) of elements from the input sets. In this case, the Cartesian product is taken over the set of actions $A_i$ for each player $i\in S$.

Combining these two concepts, the expression ${\rho }^S\in \Delta {\prod }_{i\in S}{A}_i$ means that ${\rho }^S$ is a mixed strategy profile for the set of players $S$, where each player $i\in S$ chooses a probability distribution over their set of actions $A_i$. The mixed strategy profile ${\rho }^S$ is an element of the simplex over the Cartesian product of the action sets of all the players in $S$.

The use of mixed strategies in game theory allows for the analysis of situations where players randomize their actions according to a probability distribution rather than deterministically choosing a single action. This is useful when considering equilibria in games where players have incomplete information or when considering games with multiple Nash equilibria Source 1.

In the context of the bargaining equilibrium, the mixed strategy profile ${\rho }^S$ represents the probability distributions over the actions of each player in the game. This allows for the analysis of scenarios where players don’t choose a single action with certainty but instead randomize their choices according to a probability distribution.

this doesn’t seem quite right to me. probability? seems like an LLM confabulation, perhaps? but it argues convincingly.