# Evaluating betting odds and free coupons using desirability

## TL;DR

This paper models betting odds and free coupons as desirable gambles, using the Choquet integral to identify potential for guaranteed gains for customers exploiting free coupons in the UK betting market.

## Contribution

It introduces a novel application of the Choquet integral to evaluate betting odds and free coupons, revealing how customers can achieve sure gains.

## Key findings

- Without free coupons, odds avoid sure loss.
- With free coupons, customers can identify guaranteed gain strategies.
- The approach helps assess bookmaker risk and customer advantage.

## Abstract

In the UK betting market, bookmakers often offer a free coupon to new customers. These free coupons allow the customer to place extra bets, at lower risk, in combination with the usual betting odds. We are interested in whether a customer can exploit these free coupons in order to make a sure gain, and if so, how the customer can achieve this. To answer this question, we evaluate the odds and free coupons as a set of desirable gambles for the bookmaker. We show that we can use the Choquet integral to check whether this set of desirable gambles incurs sure loss for the bookmaker, and hence, results in a sure gain for the customer. In the latter case, we also show how a customer can determine the combination of bets that make the best possible gain, based on complementary slackness. As an illustration, we look at some actual betting odds in the market and find that, without free coupons, the set of desirable gambles derived from those odds avoids sure loss. However, with free coupons, we identify some combinations of bets that customers could place in order to make a guaranteed gain.