Written during the SERI MATS program under the joint mentorship of John Wentworth, Nicholas Kees, and Janus.

Create a black and white image of a blindfolded robot and a blindfolded human playing Hungry Hungry Hippos in a 3:1 aspect ratio. The robot, with a futuristic, sleek, humanoid design, is seated at the game table with a blindfold covering its optical sensors. Across from the robot, depict a human player, also blindfolded, with an expression of concentration, as they engage in the game. The human's appearance should be neutral, representing a general human figure. The Hungry Hungry Hippos game is in the center, capturing a moment of playful competition. The background should be simple, emphasizing the players and the game in a monochromatic palette.

Motivation

I am surrounded, incessantly, by other people, possessing a range of different capabilities, striving for goals both common and opposing — and, what is more, they are incessantly surrounded by me. Fangs pierce the cartesian boundary. Outsideness leaks in. Insideness leaks out. It is the source of all joy and misery in my life.

Anyway. It's the interaction of $n \geq 2$ agents that characterises game theory.^[1] Classical game theory says that when two utility-maximisers interact in a shared environment, they will choose options in nash equilibrium, where a profile of options is in nash equilibrium if no player can increase their utility function by unilaterally changing their option.

What does higher-order game theory (which dispenses with utility functions and maximisation) say about the interaction between agents? Would a satisficer cooperate with a quantiliser in prisoner's dilemma?

Let's find out.

Preliminaries

All you need to know is Definition 3 from [Part 1], and some basic classical game theory.

Definition 3: Let $X$ be any set of options and $R$ be any set of payoffs. An optimiser is any functional $ψ : (X \to R) \to P (X)$ . A $ψ$ -task is any function $u : X \to R$ . An option $x \in X$ is $ψ$ -optimal for a task $u : X \to R$ if and only if $x \in ψ (u)$ .

Let's review the textbook definition of classical simultaneous games.

Definition 5.
Let ${X_{i}}_{i \in I}$ be a family of sets indexed by a set $I$ of players.

A classical simultaneous game over ${X_{i}}_{i \in I}$ is a function $g : \prod_{i \in I} X_{i} \to R^{I}$ .

The option-profiles of $g$ are the elements of the product $X = \prod i \in I X_{i}$ .
The $i$ -th deviation map $U_{i} : X \to (X_{i} \to X)$ is given by $U_{i} (x) (α)_{j} = {\begin{matrix} α & if i = j x_{j} & otherwise \end{matrix}$ for all $x \in X$ , $i \in I$ , $α \in X_{i}$ , and $j \in I$ .

In the finite case, where $I = {1, \dots, n}$ , then $U_{i} (x_{1}, \dots, x_{n}) : X_{i} \to X, α \mapsto (x_{1}, \dots, x_{i - 1}, α, x_{i}, \dots, x_{n})$ .

The deviation map $U_{i} : X \to (X_{i} \to X)$ describes how the $i$ th player can change an option-profile $x$ by unilaterally changing their own option. Note that the deviation maps only depend on the option spaces, not the game itself.

The best-response function of $g$ is the the function $B : X \to P (X)$ defined by $B : x \mapsto \prod i \in I {argmax}_{X_{i}} (π_{i} \circ g \circ U_{i} (x))$ where&n

...

Game Theory without Argmax