Antoine de Scorraille comments on Game Theory without Argmax [Part 1]

[Antoine de Scorraille]

Exercise 1:

The empty set is the only one. For any nonempty set X, you could pick $zer{o}_X={\lambda }^{x\in X}.0$ as a counterexample: $argma{x}_X\left(zer{o}_X\right)=X$

Exercise 2:

The agent will choose an option which scores better than the threshold $\alpha$.

It’s a generalization of satisficers, these latter are thresholders such as $u^{-1}\left(\alpha \right)$ is nonempty.

Exercise 3:

$majorit{y}_X={\lambda }^{u:X\to R}.\{x\in X|\forall x^{\prime }\in X,Card(u^{-1}\left(\right\{u\left(x\right)\left\}\right))\ge Card(u^{-1}\left(\right\{u\left(x^{\prime }\right)\left\}\right)\left)\right\}$

Exercise 4:

I have discovered a truly marvelous-but-infinite solution of this, which this finite comment is too narrow to contain.

Exercise 5:

The generalisable optimisers are the following:

$argmi{n}_X={\lambda }^{u:X\to R}.\{x\in X|\forall x^{\prime }\in X,u(x)\le u(x^{\prime })⟹u(x^{\prime })\le u(x\left)\right\}$

i.e. argmin will choose the minimal points.

$satisfic{e}_s={\lambda }^{u:X\to R}.\{x\in X|u(x)\ge u(s\left)\right\}$

i.e. satisfice will choose an option $x\in X$ which dominates some fixed anchor point $s\in X$. Note that since R is only equipped with a preorder, it means it might be a more selective optimiser (if not total, it’s harder to get an option better then the anchor). More interestingly, if there are indifferent options with the anchor (some x /​ $x\ge s$ and $s\ge x$), it could choose it rather than the anchor even if there is no gain to do so. This degree of freedom could be eventually not desirable.

$roc{k}_S={\lambda }^{u:X\to R}.S$

$satisfic{e}_S={\lambda }^{u:X\to R}.\{x\in X|\forall s\in S,u(x)\ge u(s\left)\right\}$

$majorit{y}_X={\lambda }^{u:X\to R}.\{x\in X|\forall x^{\prime }\in X,Card(u^{-1}\left(\right\{u\left(x\right)\left\}\right))\ge Card(u^{-1}\left(\right\{u\left(x^{\prime }\right)\left\}\right)\left)\right\}$

Exercise 6:

Interesting problem.

First of all, is there a way to generalize the trick?

The first idea would be to try to find some equivalent to the destested element $\perp$. For context-dependant optimiser such as better-than-average, there isn’t.

A more general way to try to generalize the trick would be the following question:

For some fixed $\psi$, $X_{legal}$ and $u:X\to R$, could we find some other $u^{\prime }:X\to R$ such as $\psi \left(u^{\prime }\right)=\psi \left(u\right)$ and $\forall x\in X_{legal},u^{\prime }\left(x\right)=u\left(x\right)$

i.e. is there a general way to replace values outside of $X_{legal}$ without modify the result of the constrained optimisation?

Answer: no. Counter-example: The optimiser $majorit{y}_X$ for some infinite set X and finite nonempty sets $X_{legal}$ and R.

So it seems there is no general trick here. But why bother? We should refocus on what we mean by constrained optimisation in general, and it has nothing to do with looking for some u’. What we mean is value outside $X_{legal}$ are totally irrelevant.

How? For any of the current example we have here, what we actually want is not ${\psi }_X\left(u^{\prime }\right)$, but ${\psi }_{X_{legal}}\left(u{|}_{X_{legal}}\right)$: we only apply the optimiser on the legal set.

Problem: in the current formalism, an optimiser has type $(X\to R)\to X$, so I don’t see obvious way to define the “same” optimiser on a different X.$argma{x}_X$ and the others here are implicitly parametrized, so it’s not that a problem, but we have to specify this. This suggests to look for categories (e.g.$Set$ for argmax...).

[rotatingpaguro]

I’d like to read your solution to exercise 6, could you add math formatting? I have a hard time reading latex code directly.

You can do that with the visual editor mode by selecting the math and using the contextual menu that appears automatically, or with $ in the Markdown editor.

There are $ in your comment, so I guess you inadvertently typed in visual mode using the Markdown syntax.

[Antoine de Scorraille]

It’s fixed ;)