Categorical Market

#Definition

A categorical market is a prediction market with three or more mutually exclusive outcomes. Traders buy shares representing their belief in each possible outcome, with prices reflecting implied probabilities that must sum to approximately 100% (or $1.00).

Unlike binary markets with just Yes/No options, categorical markets capture the full range of possibilities for multi-way events. They're common for elections with multiple candidates, tournaments with many competitors, or any question where "Which one?" replaces "Will it happen?"

#Why It Matters in Prediction Markets

Categorical markets provide richer information than binary alternatives.

Probability distribution: Rather than asking separate binary questions for each possibility, a categorical market shows relative probabilities in one view. You can see not just whether Candidate A leads, but by how much compared to B, C, and D.

Sum constraint: Because prices must sum to ~$1.00, buying one outcome implicitly sells the others. This linkage keeps prices coherent and prevents obvious arbitrage when individual binary markets on the same question would drift apart.

Tail risk visibility: Categorical markets make unlikely outcomes tradeable. A "longshot" candidate at $0.05 is clearly priced; in a series of binary markets, that option might lack enough interest to develop reliable pricing.

Information aggregation: When new information favors one candidate, prices adjust across all outcomes simultaneously, revealing not just who benefits but who loses.

#Common Categorical Markets

#How It Works

#Market Structure

A categorical market with four outcomes:

Outcome A: $0.45 (45%)
Outcome B: $0.30 (30%)
Outcome C: $0.15 (15%)
Outcome D: $0.10 (10%)
──────────────────────
Total:     $1.00 (100%)

Each outcome has its own order book or AMM position. Traders buy and sell individual outcomes like separate assets.

#Complete Set Mechanics

One share of each outcome forms a "complete set" worth exactly $1.00 at resolution:

Buy 1 share of A ($0.45)
Buy 1 share of B ($0.30)
Buy 1 share of C ($0.15)
Buy 1 share of D ($0.10)
────────────────────────
Total cost: $1.00
Guaranteed payout: $1.00 (one outcome wins, pays $1)

This complete set property enforces the sum constraint. If prices summed to more than $1.00, traders could short the complete set for risk-free profit.

/**
 * Normalizes a set of categorical probabilities to sum to 1.0.
 * Useful when raw model outputs don't sum to 100%.
 * 
 * @param probabilities - Array of raw probabilities (e.g., [0.4, 0.3, 0.2])
 * @returns Array of normalized probabilities
 */
function normalizeProbabilities(probabilities) {
  const sum = probabilities.reduce((a, b) => a + b, 0);
  
  if (sum === 0) return probabilities; // Avoid division by zero
  
  return probabilities.map(p => parseFloat((p / sum).toFixed(4)));
}

// Example: Model outputs [0.50, 0.40, 0.30] -> Sum 1.20
const normalized = normalizeProbabilities([0.50, 0.40, 0.30]);
// Result: [0.4167, 0.3333, 0.2500] -> Sum ~1.0

#Numerical Example

A market asks: "Which candidate will win the election?"

Current prices:

• Candidate A: $0.52
• Candidate B: $0.35
• Candidate C: $0.08
• Others: $0.05

Your analysis suggests:

• Candidate A: 45% probability (market says 52%)
• Candidate B: 42% probability (market says 35%)

Trade: Sell A shares at $0.52, buy B shares at$0.35

If B wins:

B payout: $1.00 per share → profit $0.65 per share
A position: expires worthless → keep the $0.52 received
Net profit: $0.65 + $0.52 - $0.35 = $0.82 per pair

If A wins:

A pays $1.00 → you owe $0.48 (sold at $0.52)
B expires worthless → lose $0.35
Net loss: $0.48 + $0.35 = $0.83 per pair

The trade is profitable if B's true probability exceeds the market's implied 35%.

#Examples

#Example 1: Primary Election

A party primary market with five candidates:

• Candidate Alpha: $0.40
• Candidate Beta: $0.25
• Candidate Gamma: $0.18
• Candidate Delta: $0.12
• Field (all others): $0.05

The market shows Alpha as frontrunner but not dominant. Traders see value in Beta or Gamma if momentum shifts. The Field category captures longshot possibilities.

#Example 2: Federal Reserve Decision

A categorical market on the Fed's rate action:

• Cut 50 bps: $0.05
• Cut 25 bps: $0.15
• No change: $0.55
• Hike 25 bps: $0.20
• Hike 50 bps: $0.05

This structure captures the full probability distribution, not just "will they cut?" A trader who expects 35% chance of any hike can buy both hike outcomes for $0.25.

#Example 3: Championship Winner

A sports league market with 30 teams:

• Team Alpha: $0.18
• Team Beta: $0.12
• Team Gamma: $0.10
• ...
• Team Omega: $0.01

Deep categorical markets can have dozens of outcomes.

#Scalar Buckets

Sometimes, a Scalar Market (e.g., "What will the temperature be?") is implemented as a Categorical Market using "buckets":

• Bucket A: < 50°F
• Bucket B: 50-70°F
• Bucket C: > 70°F This simplifies trading by turning a continuous range into discrete, mutually exclusive options.

#Example 4: Oscar Nominations

A market on "Best Picture winner" with 10 nominees:

Each film's price represents its winning probability. Late-breaking reviews, box office performance, or guild awards shift probabilities across all nominees as the ceremony approaches.

#Risks and Common Mistakes

Ignoring liquidity fragmentation

Liquidity spreads across all outcomes. A categorical market with 10 options at $100,000 total depth has only$10,000 average depth per outcome, far less than a single binary market with $100,000 depth.

Sum drift from spreads

The sum of mid-prices may drift from $1.00 due to bid-ask spreads. If all outcomes show ask prices, the sum exceeds$1.00; if all show bids, it falls below. This isn't arbitrage; it reflects execution costs.

Correlation errors

News often affects multiple outcomes. A scandal hurting Candidate A benefits all alternatives, not just the second-place candidate. Failing to update across correlated outcomes leads to inconsistent positions.

Overweighting longshots

Prices near $0.05 can feel like "cheap lottery tickets." But buying a$0.05 outcome expecting 10% probability only yields 2:1 expected edge, while most of the time you lose everything. The numbers often look better than the reality.

Missing resolution rules

Categorical markets must handle edge cases: what if a candidate withdraws? What if two outcomes tie? Resolution rules vary by platform and market; read them carefully.

#Practical Tips for Traders

• Compare against binary alternatives: If binary markets exist for individual outcomes, check for pricing discrepancies with the categorical market

• Use complete sets for arbitrage detection: If the sum of ask prices falls below $1.00, you can buy the complete set below guaranteed payout (rare but possible)

• Focus on relative value: Instead of asking "Is A overpriced?", ask "Is A overpriced relative to B?" Relative trades reduce exposure to overall probability misjudgment

• Layer across outcomes: Rather than betting everything on one outcome, spread positions across the 2-3 outcomes you find most underpriced

• Check resolution for edge cases: Understand what happens if your candidate withdraws, if there's a tie, or if the event is postponed

• Mind the tail liquidity: Outcomes below $0.10 often have very wide spreads. The displayed price may not reflect executable levels

• Calculate implied probabilities explicitly: Prices are probabilities. A $0.25 outcome needs to happen 25% of the time just to break even.

#The Math of Categorical Markets

In a categorical market, since exactly one outcome must occur, the probabilities of all outcomes must sum to 100% (or $1.00).

$P(A) + P(B) + P(C) = 1.00$

Example: "Who will win the election?"

• Candidate A: $0.40
• Candidate B: $0.35
• Candidate C: $0.25
• Total: $1.00

If the total price is **less than $1.00** (e.g.,$0.90), you can buy YES on all outcomes and guarantee a profit (buy A+B+C for $0.90, get back$1.00). If the total price is **more than $1.00** (e.g.,$1.10), you can sell YES (or buy NO) on all outcomes to profit.

#Related Terms

• Binary Market
• Scalar Market
• Implied Probability
• Arbitrage
• Order Book
• Liquidity

#FAQ

#How does a categorical market differ from a binary market?

A binary market has exactly two outcomes (Yes/No) that sum to $1.00. A categorical market has three or more mutually exclusive outcomes that together sum to$1.00. Binary markets are simpler but categorical markets capture multi-way probability distributions.

#What happens if prices don't sum to exactly $1.00?

Small deviations are normal due to bid-ask spreads and timing. If the sum significantly exceeds $1.00 using ask prices, it reflects the cost of buying a complete set. If the sum falls significantly below$1.00 using bid prices, selling a complete set loses money. True pricing arbitrage (risk-free profit) is rare on active platforms.

#Can I trade multiple outcomes at once?

Yes. Many traders take relative positions: selling one outcome while buying another. This "pairs trade" profits if the relative probability shifts in your favor, partially hedging against movements affecting all outcomes equally.

#Are categorical markets harder to trade than binary markets?

They require more attention to interdependencies: buying one outcome affects your effective position on all others. Liquidity fragmentation across outcomes can also make execution harder. However, the additional structure can reveal opportunities that binary markets obscure.

#How do categorical markets resolve when a candidate withdraws?

Platform rules vary. Some markets resolve remaining candidates proportionally, some return funds to the withdrawn outcome's holders, and some have explicit "withdrawal means No" rules. Always check the specific resolution criteria before trading.