Exercises: Chapter 18

Conceptual Understanding

Exercise 1: Bias Identification

For each of the following scenarios, identify the primary behavioral bias at work and explain how it leads to mispricing:

a) A prediction market contract on "Will a Category 5 hurricane hit the U.S. mainland this year?" is trading at 18% in September, one week after a Category 4 hurricane made landfall. Historical base rate for Category 5 landfalls in any given year is approximately 3%.

b) A new contract opens on "Will the Federal Reserve raise interest rates at the next meeting?" The first trade occurs at $0.50, and after extensive analysis by multiple traders, the price has moved to $0.53 over two days. An economic model with a strong track record estimates the probability at 72%.

c) After a candidate wins three primary elections in a row by large margins, traders price their general election victory at 78%, even though the candidate's party has lost four of the last five general elections and current head-to-head polling shows a tied race.

d) A trader who bought YES shares on a contract at $0.65 refuses to sell at $0.40, saying "I'll wait for it to come back," even though they now privately believe the true probability is around 35%.

Exercise 2: Favorite-Longshot Bias Calculation

You have the following data from 5,000 resolved prediction market contracts:

a) Calculate the deviation (actual - implied midpoint) for each bin. b) Is the favorite-longshot bias present? How do you know? c) Estimate the parameter $\gamma$ of the prospect theory weighting function that best fits this data. d) If you could only trade in one probability bin, which would offer the highest expected profit per contract?

Exercise 3: Anchoring Quantification

A prediction market contract for "Will Candidate X win the election?" is currently trading at $0.52. The most recent major poll shows Candidate X with 55% support. Before the poll was released, the market was trading at $0.48.

a) If traders are anchoring to the pre-poll price with an adjustment coefficient of $\alpha = 0.5$, and the poll is a perfect signal of the true probability, what is the expected mispricing? b) If you believe $\alpha = 0.5$ and the true probability is 55%, what trade would you make and what is your expected profit per share? c) What factors might cause $\alpha$ to be higher (closer to 1) or lower (closer to 0)?

Exercise 4: Overconfidence Assessment

A trader has made 200 predictions on binary events. For predictions where they assigned a probability of 80% or higher, the actual outcome matched their prediction only 65% of the time. For predictions where they assigned 50-60%, outcomes matched 52% of the time.

a) Calculate the calibration error for each probability range. b) Is this trader overconfident, underconfident, or well-calibrated? c) How would you advise this trader to adjust their probability estimates? d) If this trader's miscalibration is typical of the market, what trading strategy would you recommend?

Exercise 5: Herd Behavior Analysis

The following price and volume data is observed for a prediction market contract over 10 consecutive time periods. No significant news events occurred during this period.

a) Calculate the return for each period. b) Does this data show signs of herding? What specific features suggest herding? c) At what point would a contrarian trader have entered a position, and in which direction? d) What was the contrarian trader's profit or loss per share if they entered at the optimal point?

Applied Analysis

Exercise 6: Availability Bias in Pandemic Markets

After a novel respiratory virus is detected in Southeast Asia, a prediction market contract on "Will the WHO declare a Public Health Emergency of International Concern within 90 days?" jumps from $0.05 to $0.25 in one day, on heavy volume. The base rate for PHEIC declarations is approximately one every 3-4 years.

a) Calculate the base rate probability of a PHEIC declaration in any given 90-day window. b) How much of the price jump is likely explained by the new information (virus detection) versus availability bias (memory of COVID-19)? c) Design a trading strategy that accounts for both the new information and the availability bias. d) What additional information would you need to refine your probability estimate?

Exercise 7: Narrative Fallacy Case Study

Consider the following timeline for a prediction market on a tech company's product launch:

• Day 1: Contract listed at $0.50 for "Will Company X launch Product Y by December 31?"
• Day 15: Tech blog reports "sources say development is ahead of schedule." Price rises to $0.62.
• Day 30: Company CEO tweets enthusiastically about Product Y. Price rises to $0.70.
• Day 45: Industry analyst predicts Product Y will "change everything." Price rises to $0.75.
• Day 60: A key engineer leaves Company X. Price barely moves, staying at $0.74.
• Day 75: Company announces a partnership related to Product Y. Price rises to $0.80.

a) Identify the narrative being constructed. b) At which points might the narrative be causing mispricing? c) The engineer departure (Day 60) is potentially negative information. Why did the market barely react? d) If the base rate for on-time product launches by this company is 40%, and none of the narrative events above would meaningfully change this base rate in a Bayesian analysis, estimate the mispricing at Day 75.

Exercise 8: Disposition Effect Simulation

A trader buys 100 shares of contract A at $0.60 and 100 shares of contract B at $0.40. After one week: - Contract A is trading at $0.70 (10-cent gain) - Contract B is trading at $0.30 (10-cent loss)

The trader now believes contract A's fair value is $0.75 (5 cents above market) and contract B's fair value is $0.25 (5 cents below market).

a) What would a rational, loss-neutral trader do? b) What does prospect theory predict the trader will do? c) Calculate the expected cost of the disposition effect if the trader sells A and holds B, versus the rational action. d) Design a rule that would prevent the disposition effect in this scenario.

Exercise 9: Confirmation Bias Audit

You hold YES shares on a contract at $0.65. Over the past week, you have encountered the following pieces of evidence:

1. A supportive analysis from an analyst you respect (+)
2. A negative data point from a reliable source (-)
3. A social media thread arguing for YES (+)
4. A detailed statistical model showing the probability at $0.55 (-)
5. A friend who agrees with your position (+)
6. A prediction market community poll showing 70% YES (+)

a) Characterize each piece of evidence as "strong" or "weak" and "confirming" or "disconfirming." b) If you were perfectly rational, how would you weight each piece of evidence? c) How does confirmation bias distort your weighting? d) After properly weighting all evidence, should you adjust your position? In which direction?

Exercise 10: Combined Bias Detection

A contract on "Will Country X default on its debt in the next year?" is trading at $0.15. You observe the following: - The historical base rate for sovereign default is approximately 2% per country-year. - Country X recently experienced a high-profile political crisis that was extensively covered in the media. - A popular prediction market commentator recently published a thread arguing that default is likely. - The price rose from $0.08 to $0.15 over three days with increasing volume and no new economic data. - The contract price has been hovering near $0.15 (a round number) for the past day.

List every bias that might be contributing to the current price of $0.15, quantify the approximate contribution of each, and estimate the fair value of the contract.

Quantitative Problems

Exercise 11: Prospect Theory Value Function

Using the standard prospect theory parameters ($\alpha = 0.88$, $\beta = 0.88$, $\lambda = 2.25$):

a) Calculate the prospect theory value of gaining $50. b) Calculate the prospect theory value of losing $50. c) What size gain would produce the same magnitude of psychological value as a $50 loss? d) A trader faces a choice: a certain gain of $30 or a 50% chance of gaining $70. Under expected utility with a risk-neutral utility function, which is preferred? Under prospect theory, which is preferred? Show your work.

Exercise 12: Kelly Criterion with Bias Correction

You detect a favorite-longshot bias in a market. The contract is trading at $0.08 (implying 8% probability). Your analysis, accounting for the FLB, suggests the true probability is 5%.

a) Calculate the Kelly fraction for a SHORT position (selling at $0.08 when you believe the true probability is 5%). b) If your bankroll is $10,000, what position size does Kelly recommend? c) If you are uncertain about your bias estimate (you think the true probability could be anywhere from 3% to 7%), how should you adjust the Kelly fraction? d) What is the expected profit per contract from this trade, before transaction costs?

Exercise 13: Information Cascade Model

Consider a sequential trading model where each trader receives a private binary signal (H = event will happen, L = event will not happen) with accuracy 60% (i.e., P(H | event happens) = 0.6 and P(L | event doesn't happen) = 0.6). The prior probability of the event is 50%.

a) Trader 1 receives signal H and buys. What is the posterior probability after Trader 1's action? b) Trader 2 receives signal L. Given that Trader 1 bought (revealing signal H), what is Trader 2's posterior? Does Trader 2 buy or sell? c) Trader 3 receives signal L. Given that Traders 1 and 2 both bought, what is Trader 3's posterior? Does Trader 3 buy or sell? d) At what point does an information cascade form (where traders ignore their own signal)?

Exercise 14: Calibration Curve Analysis

A trader's calibration data over 500 predictions:

a) Calculate the mean calibration error. b) Plot (or describe) the calibration curve. Is this trader overconfident or underconfident? c) For which probability ranges is the miscalibration most severe? d) Design a recalibration function that would correct this trader's estimates.

Exercise 15: Herding Detection Statistics

You have the following return autocorrelation data for a prediction market:

• Lag 1 autocorrelation: +0.25
• Lag 2 autocorrelation: +0.15
• Lag 5 autocorrelation: -0.10
• Lag 10 autocorrelation: -0.20

a) Interpret these autocorrelation values. What pattern do they suggest? b) Is this pattern consistent with herding? Explain. c) A purely informationally efficient market would have autocorrelations close to zero at all lags. Calculate the "herding index" as the difference between the average short-term autocorrelation (lags 1-2) and the average medium-term autocorrelation (lags 5-10). d) If you trade a mean-reversion strategy based on this autocorrelation structure, what is the optimal holding period?

Strategy Design

Exercise 16: FLB Exploitation Strategy

Design a complete trading strategy that exploits the favorite-longshot bias. Your strategy should include:

a) Entry criteria: At what price levels do you enter positions? Buy or sell? b) Position sizing: How do you size positions relative to the estimated bias? c) Exit criteria: When do you close positions? d) Risk management: How do you handle the risk that the bias may not be present in a specific contract? e) Write pseudocode for the strategy.

Exercise 17: Contrarian Herding Strategy

Design a contrarian strategy that profits from herding-driven price movements.

a) How do you identify herding episodes in real-time? b) What signals trigger entry? c) How do you distinguish herding from genuine information flow? d) What is your expected holding period? e) What is the biggest risk of this strategy, and how do you mitigate it?

Exercise 18: Multi-Bias Signal Combiner

You have access to the following bias signals for a single contract: - FLB signal: +0.03 (longshot overpricing detected) - Anchoring signal: +0.02 (price stuck above fundamental) - Herding signal: +0.04 (reversal expected) - Availability signal: +0.03 (vivid event overweighting) - Recency signal: -0.01 (recent information suggests underpricing)

a) Calculate the combined signal using equal weights. b) Calculate the combined signal using the weights from the BiasExploitationFramework in Section 18.10. c) Should you trade? If so, in which direction and with what confidence? d) What additional information would strengthen or weaken the combined signal?

Exercise 19: Debiasing Protocol

Design a complete debiasing protocol for a prediction market trader. Include:

a) A pre-trade checklist with at least 8 items. b) A post-trade review template. c) A weekly self-assessment framework. d) Metrics for tracking improvement over time.

Exercise 20: Prospect Theory Position Management

A trader using prospect theory value maximization (rather than expected value) holds a position purchased at $0.50. The current price is $0.55 (in profit). The trader estimates: - 40% chance the price goes to $0.65 (further gain) - 30% chance the price stays at $0.55 (no change) - 30% chance the price drops to $0.45 (turns to loss)

a) Calculate the expected value of holding versus selling now. b) Calculate the prospect theory value of holding versus selling now (use $\alpha = 0.88, \lambda = 2.25$). c) Under expected value, should the trader hold or sell? Under prospect theory? d) If the trader is currently at a loss (bought at $0.60, current price $0.55), redo the calculation. How does the answer change?

Advanced Problems

Exercise 21: Bayesian vs. Biased Updating

A contract is at $0.50. A new poll is released showing 55% support for the event. A perfectly Bayesian trader would update the probability to $0.53 (given the poll's precision and other information). A recency-biased trader overweights the poll and updates to $0.56.

a) Calculate the mispricing if the market is dominated by recency-biased traders. b) Over 100 such poll releases, what is the expected cumulative profit of a trader who always trades toward the Bayesian estimate? c) What is the variance of this strategy's returns? d) Calculate the Sharpe ratio.

Exercise 22: Information Cascade Fragility

In an information cascade, all traders are following the crowd. But cascades are "fragile" — a single piece of strong public information can break them.

a) In the model from Exercise 13, suppose the cascade has formed in the wrong direction (the event will not happen, but traders are all buying). What signal strength is needed for a single piece of public information to break the cascade? b) How does the fragility of cascades create trading opportunities? c) Design a strategy that profits from cascade breaks. What data would you need?

Exercise 23: Cross-Bias Interaction

Some biases reinforce each other, while others counteract each other. For each pair below, explain whether they reinforce or counteract, and give an example:

a) Availability bias + Recency bias b) Anchoring + Herding c) Overconfidence + Favorite-longshot bias d) Confirmation bias + Loss aversion e) Narrative fallacy + Representativeness heuristic

Exercise 24: Bias Decay Over Time

Some biases are stronger far from resolution and weaker close to resolution (as uncertainty resolves). Others persist until the end.

a) For each major bias (FLB, anchoring, overconfidence, herding, availability), predict whether it gets stronger, weaker, or stays constant as resolution approaches. b) Design a time-varying bias exploitation strategy that adjusts weights based on time to resolution. c) What implications does bias decay have for position timing (when to enter and exit)?

Exercise 25: Market Microstructure and Bias

The structure of a prediction market can amplify or dampen behavioral biases.

a) How does a continuous double auction (order book) versus an automated market maker (AMM) affect the favorite-longshot bias? b) How do position limits affect herding? c) How does the presence or absence of short-selling affect the disposition effect? d) Design a market mechanism that would reduce the impact of behavioral biases.

Programming Exercises

Exercise 26: Build a Calibration Tracker

Write a Python program that: a) Accepts probability predictions and outcomes. b) Computes calibration curves and Brier scores. c) Identifies the probability ranges where the user is most miscalibrated. d) Suggests corrections based on historical miscalibration.

Exercise 27: Simulate the Favorite-Longshot Bias

Write a Python simulation that: a) Generates 10,000 events with true probabilities drawn uniformly from [0, 1]. b) Generates market prices by applying prospect theory probability weighting to the true probabilities. c) Adds noise to the market prices. d) Runs the FLB detection algorithm from Section 18.2 and verifies it detects the bias. e) Simulates a contrarian strategy and computes its profitability.

Exercise 28: Herding Simulation

Write a Python simulation of an information cascade: a) Model 50 sequential traders, each receiving a private signal. b) Track the market price as each trader acts. c) Identify when cascades form and measure their duration. d) Inject a "public signal" at random points and measure how often it breaks the cascade.

Exercise 29: Decision Journal Analyzer

Write a Python program that: a) Reads a CSV file of trading decisions (date, contract, predicted probability, market price, confidence, reasoning, outcome). b) Computes all bias metrics from this chapter (calibration error, disposition effect, anchoring to round numbers, overconfidence). c) Generates a summary report with specific recommendations.

Exercise 30: Complete Bias Exploitation Backtest

Using the BiasExploitationFramework from Section 18.10: a) Generate a synthetic dataset of 5,000 resolved markets with realistic bias patterns. b) Run the framework's backtest method. c) Vary the bias weights and measure the impact on performance. d) Find the optimal weights using a simple grid search. e) Test whether the optimal weights are stable across different subsets of the data (cross-validation).