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nash equilibrium

a nash equilibrium is a set of strategies, one per player, where nobody can do better by changing only their own move. each player is already playing a best response to everyone else, so nobody has a reason to deviate.

it needn't be the outcome everyone prefers. it is only the outcome no one can escape alone. that gap, between what's stable and what's good, is most of what makes game theory interesting.

the classic picture is two suspects deciding whether to confess. each confessing is an equilibrium even though both would do better staying quiet, because neither can improve by changing only their own move.

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best response

you asked: what does this mean here?

the move that does best for you given what everyone else is doing. an equilibrium is just the point where every player is playing one at the same time.

if a move is your best response no matter what the others do, it's a dominant strategy. most games don't have one, which is why equilibrium is the more useful idea.

dominant strategy

a move that beats your alternatives whatever the others choose. in the suspects' game, confessing is dominant for both, which is why the bad outcome is stable.

when every player has one, the equilibrium is easy to find. when nobody does, you need the full best-response reasoning, and it is not obvious an equilibrium exists at all.

that's the gist. this thread has what you came for. ↩ back

existence

nash showed every finite game has at least one equilibrium if players may randomise. the proof leans on a fixed-point theorem.

drifting from your goal. the ideas, not the proofs.

fixed point

a point a function maps to itself. kakutani's version handles set-valued maps, which is what best responses are.

further from the goal. shorter from here on.

kakutani

a fixed-point theorem for upper hemicontinuous correspondences on compact convex sets.

parked as a seed →

hemicontinuity

a continuity condition on set-valued maps.

off goal. parked as a seed →

scroll sideways. the path reads left to right, and each pane past the goal says less than the one before it.

the wall

you hit a word you don't know. asking means a new tab and a lost place, so you skip it and tell yourself you got the gist. that flinch is why reading stays shallow.

here, the word becomes a pane. the one you left stays where it was.

the goal

you ramble what you want. a short interview pins it: the topics, how far in to go, what you want to be able to say at the end, and what sits just outside.

nash equilibrium best response dominant strategies existence proofs topology

the rules

it never blocks you. panes get shorter and dimmer. that's all.
every claim has a source you can open.
it won't connect the dots for you. that's your job.
it asks, it never grades. no scores, no streaks, no queues.
it's not a notes app. understand here, write notes elsewhere.

why

i have adhd. i take notes instead of learning and collect breadth instead of depth. this is the tool i wanted.

superficial skill is about to be free. depth is what's left.

— sai