The Topology of Choice
A Small Tree, a Hidden State and the Price of Knowing
Canonical essay: https://ludologium.com/monographs/topology-of-choice
Editorial essay by Evelyna Bellamy; not externally peer reviewed. 692 words. Estimated at 200 words per minute for the abstract and main prose; references and interactive studies take additional time. Publication details revised 10 October 2026.
Revised 2026-10-11. Editorial essay with explicit model, source and measurement limits.
Abstract and scope
A game tree describes what can happen; an information set describes what a player can know when choosing. Two small constructed examples make that distinction inspectable and show why a search result depends on declared payoffs, observations and opponent assumptions. No equilibrium or reinforcement-learning result is claimed.
1. Count the choices before praising the search
Imagine a player at a fork. Left leads to a gatekeeper who chooses between payoffs 3 and 1. Right leads to a gatekeeper who chooses between 2 and 0. The numbers are terminal utilities for the first player, not currency, probability or observed satisfaction. Both players see the entire tree. The gatekeeper wants to minimize that utility; the first player wants to maximize it. These assumptions define a constructed two-player zero-sum example.
Work backward. The gatekeeper selects 1 after Left and 0 after Right. The first player therefore selects Left for a guaranteed utility of 1 under the stated assumptions. Choosing the largest visible leaf, 3, would give the same initial direction here for the wrong reason; change Left to leaves 9 and −2, and that shortcut would select the branch whose guaranteed outcome is −2 rather than 0.
The result is small enough to verify on paper. It demonstrates a decision rule, not intelligent play in every game. If the gatekeeper prefers cooperation, makes errors or has a different payoff, the minimization step no longer describes that opponent. A search algorithm does not supply missing preferences.
2. Hide a state, change the question
Now replace the gatekeeper with a hidden token drawn before the player chooses. The token is Red or Blue, each with stipulated probability one half. Choosing Left pays 4 for Red and −2 for Blue. Choosing Right pays 1 regardless of color. The player knows the distribution and payoffs but cannot observe the token. This is another constructed game, not a poker model.
The two possible decision nodes belong to one information set: the player must choose the same action distribution in both because the states are indistinguishable at that moment. Without that constraint, an analyst could choose Left only in the Red state and Right only in Blue, giving the player knowledge the game never supplied.
Expected utility for Left is (4 + −2)/2 = 1; for Right it is 1. A risk-neutral player is indifferent under those probabilities. A player whose objective is the worst possible payoff chooses Right, whose minimum is 1 rather than −2. These are different objectives, not competing arithmetic answers to one fully specified question.
3. Information has a value within the model
If the token is revealed before the choice, selecting Left on Red and Right on Blue yields expected utility (4 + 1)/2 = 2.5. Under the risk-neutral assumptions, the gross value of that information is 1.5 compared with the unrevealed best expected utility of 1. If revelation costs more than 1.5 utility units, buying it would reduce expected utility. Nothing here says information is always worth its price.
This comparison isolates the order of observation and action. A cue that appears after a choice cannot be credited as if it appeared before. A strategy is therefore a plan for what the player will do at the information available to them, rather than a list of ideal actions selected with hindsight.
4. The boundary of the tree
Shannon’s 1950 chess paper offers a historical source for programmatic search and evaluation. The archive’s former numerical shorthand mixed an approximate branching calculation with claims that other methods became obsolete. Those claims are withdrawn. A uniform tree with branching factor b and depth d has b^d leaves; an actual game can have unequal branching, repeated states, early endings and expensive evaluation.
The useful design question is which choices and observations the model leaves out. Does a player know an opponent’s action? Are payoffs comparable? What happens when the horizon ends? Can a tree represent a simultaneous choice only by hiding an earlier move? Each answer changes what a computed recommendation means.
I would publish the tree, information sets, chance probabilities and utility convention beside the result. A reader should be able to reproduce the small example before trusting a large search claim. The topology of choice becomes meaningful when it tells us both where a player can go and what they are permitted to know on the way.