Long-Shot Bias

#Definition

Long-Shot Bias (or Favorite-Longshot Bias) is one of the most documented anomalies in betting markets: traders systematically overvalue low-probability outcomes ("long shots") while undervaluing favorites.

First documented by Griffith in 1949 in horse racing, this bias has been observed repeatedly across sports betting, prediction markets, and financial options markets. Traders behave as if a 1% chance is actually 5% or 10%—paying a premium for "lottery ticket" payoffs that violates rational expected value calculations.

#Empirical Evidence

The bias is remarkably consistent across decades of research:

Research by Snowberg and Wolfers (2010) found that betting on horses at 100:1 odds or longer yields average returns of -61%, while betting randomly returns -23%, and betting favorites performs significantly better.

#Why It Matters

• Negative EV Trap: Long shots are systematically overpriced. The market pays 50-to-1 on what should be 100-to-1 odds.

• Persistent Inefficiency: Despite being well-documented for 75+ years, the bias persists—creating opportunities for sophisticated traders who sell (short) long shots.

• Geographic Variation: Interestingly, Asian markets (Hong Kong, Japan) sometimes show a reversed bias, where favorites are overbet. This suggests cultural and structural factors play a role.

#How It Works

#Prospect Theory & Probability Weighting

Humans weight probabilities non-linearly. We overweight small probabilities (fear/hope) and underweight large probabilities (certainty effect).

#Python: The Bias Function

This script simulates how humans distort low probabilities using the Prelec weighting function (common in behavioral econ).

import math

def probability_weight(p, alpha=0.65):
    """
    Prelec function: w(p) = exp(-(-ln(p))^alpha)
    Alpha < 1 implies overweighting low probs and underweighting high probs.
    """
    if p == 0: return 0
    if p == 1: return 1
    
    w_p = math.exp(-((-math.log(p))**alpha))
    return w_p

# Example: True probability 1% vs 99%
true_low = 0.01
true_high = 0.99

biased_low = probability_weight(true_low)
biased_high = probability_weight(true_high)

print(f"True Low: {true_low:.1%} -> Perceived: {biased_low:.1%}")
print(f"True High: {true_high:.1%} -> Perceived: {biased_high:.1%}")

# Output implies:
# Traders will pay ~5-6 cents for a 1 cent contract.
# Traders will only pay ~95 cents for a 99 cent contract.

#Competing Explanations

Researchers debate why the bias persists:

1. Risk-Love (Neoclassical): Gamblers rationally overpay for long shots because they enjoy the thrill of high-variance bets

2. Probability Misperception (Behavioral): Humans systematically overweight small probabilities due to cognitive biases (prospect theory)

Snowberg and Wolfers (2010) tested these explanations using compound bets (exactas, trifectas) and found stronger evidence for misperception—people genuinely believe long shots win more often than they do.

#Examples

#Horse Racing

Horses with 2% implied odds typically have actual win rates around 0.7%—generating ROI of -40% or worse. Meanwhile, favorites (50% implied odds) return approximately -5% (just the house take).

#Prediction Markets

The "Lottery Ticket" Trap on Polymarket:

• A candidate polling at 1% often trades at 4-5 cents
• Casual traders think: "Anything can happen! Only $0.04 for a potential$1.00 payout!"
• Reality: Buying at $0.04 implies 1-in-25 odds, but true odds may be 1-in-200
• The seller pockets the premium systematically

Real Example: "Will Trump say 'Crypto' during a Bill Signing?" trading at 30 cents. Data-driven traders analyzed his last 10 speeches (never used the word), bought NO at 70 cents, and profited when the contract resolved to NO.

#The Scale of Opportunity

From April 2024 to April 2025, arbitrageurs extracted over **$40 million** from Polymarket by exploiting systematic mispricings—including the long-shot bias. The top three wallets alone earned$4.2 million. Yet only 0.51% of wallets achieved profits exceeding $1,000.

#Risks and Common Mistakes

Buying "Cheap" Contracts: Novice traders love $0.02 shares. "50x potential!" They ignore that hitting$1.00 requires a near-impossible event. Over thousands of such bets, you lose 60%+ of your capital.

Shorting Dangers: Profiting from this bias means selling (shorting) the long shot. But if you short a $0.02 contract and it spikes to$0.15 on news or manipulation, you face 650% loss on that position—even if it eventually resolves to $0.00.

Illiquidity: Long-shot markets are often thin. You might profit on paper but be unable to exit your position at a reasonable price.

#Trading Strategies

For Sophisticated Traders:

• Systematically sell long shots, accepting high variance for positive expected value
• Use position limits to survive temporary spikes
• Focus on markets with clear resolution criteria

For Everyone Else:

• Avoid the temptation of "cheap" contracts
• If you must speculate, size positions assuming total loss
• Betting exchanges show smaller biases than traditional bookmakers—prefer them