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Decision markets

A decision market turns action-conditional forecasts into a choice. For each candidate action $a$ it runs a conditional market that predicts an outcome given that $a$ is taken; a decision rule maps the resulting per-action value estimates $v_a = \mathbb{E}[\text{value}\mid a]$ to a choice; and only the chosen action's market settles against what actually happens. It is best read as an ensembling-and-selection mechanism: each conditional market is an aggregate of its traders' beliefs, and the decision market ensembles across those action-conditional aggregates and picks one.

Conditional markets, one per action

Stand up one LMSR per action over the shared outcome space (combinatorial markets supply the same conditional machinery when actions and outcomes share variables). Reading off each market's prices gives a matrix of conditional distributions, and dotting each row against a value vector over outcomes gives the per-action value:

In code, DecisionMarket(n_actions, n_outcomes) holds the markets; trade(action, outcome, shares) moves one conditional; values(outcome_values) returns the vector $v$; and conditional_matrix() returns the full $(A \times K)$ table.

From conditional prices to a choice

A decision rule maps the value vector $v$ to a distribution over actions. The obvious one is deterministic argmax: put all weight on the best action. The softer alternatives keep some weight everywhere: the Boltzmann (softmax) rule $P(a) \propto e^{v_a/T}$, which sharpens toward argmax as the temperature $T \to 0$, and $\varepsilon$-greedy, which plays the argmax with probability $1-\varepsilon$ and spreads $\varepsilon$ uniformly over all actions.

Why the rule must be stochastic

Here is the subtlety that makes a decision market more than a pair of markets side by side. Under a deterministic argmax rule the conditional markets are not incentive compatible: a trader can move which action is selected, and the markets for the actions not taken never settle, so reports there are never scored and can be pushed around for free. The fix is a decision rule with full support: if every action keeps a positive probability of being chosen, every conditional market settles with positive probability, and a risk-neutral trader maximises expected payoff by reporting its true conditional belief in each one. Softmax and $\varepsilon$-greedy have full support; argmax does not (Othman & Sandholm 2010; Chen, Kash, Ruberry & Shnayder 2011). The governance reading, "vote on values, bet on beliefs" (Hanson 2013), sits on top of this market core and is left out as social theory.

Try it

Three actions, each with a conditional market predicting the probability of a good outcome (value 1). Set those conditionals, pick a decision rule, and watch the choice distribution. Switch to argmax and an action's bar collapses to zero; its market goes unsettled, the manipulable case the stochastic rules repair.

Code: mechanisms/decision_market.py; demo examples/sim_decision_market.py · Related: combinatorial & conditional markets, LMSR, aggregation · Research: gaps-and-roadmap.md