# The Doping Game: Payoffs That Make Cheaters Into Losers

Editors note: This story is part of a Feature "The Doping Dilemma" from the April 2008 issue of Scientific American.

Why do cyclists cheat? The game theory analysis of doping in cycling (below), which is closely modeled on the game of prisoner’s dilemma, shows why cheating by doping is rational, based solely on the incentives and expected values of the payoffs built into current competition. (The expected value is the value of a successful outcome multiplied by the probability of achieving that outcome.) The payoffs assumed are not unrealistic, but they are given only for illustration; the labels “high,”  “temptation,” “sucker” and “low” in the matrices correspond to the standard names of strategies in prisoner’s dilemma. It is also assumed that if competitors are playing “on a level playing field” (all are cheating, or all are rule-abiding), their winnings will total \$1 million each, without further adjustment for a doping advantage. 

—Peter Brown, Staff Editor

Game Assumptions: Current Competition

• Value of winning the Tour de France: \$10 million
• Likelihood that a doping rider will win the Tour de France against nondoping competitors: 100%
• Value of cycling professionally for a year, when the playing field is level: \$1 million
• Cost of getting caught cheating (penalties and lost income): \$1 million
• Likelihood of getting caught cheating: 10%
• Cost of getting cut from a team (forgone earnings and loss of status): \$1 million
• Likelihood that a nondoping rider will get cut from a team for being noncompetitive: 50%

 Case I: My opponent abides by the rules (he "cooperates"). I have two options: Case 2: My opponent cheats by doping (he "defects"). Again, I have two options: High Payoff Sucker Payoff I abide by the rules (I "cooperate," too). The playing field is level. I abide by the rules (I "cooperate"). I can earn the average winnings for a competitive racer only if my opponent gets caught cheating and is disqualified. Value of competing for one year: \$1 million Expected value of competing for one year: \$1 million * 10% = \$0.1 million Since I am not cheating, I expect no penalties: \$0 Expected cost of getting cut from a team: \$1 million * 50% = -\$0.5 million Total expected High Payoff: \$1 million Total expected Sucker Payoff: \$0.4 million – – Temptation Payoff Low Payoff I cheat by doping (I "defect"). I also cheat by doping (I "defect"). The playing field is level. Expected value of winning the Tour de France (if I do not get caught cheating): \$10 million * 90% = \$9.0 million Expected value of competing for one year (if I do not get caught): \$1 million * 90% \$0.9 million Expected penalty for cheating (if I do get caught): \$1 million * 10% = -\$0.1 million Expected penalty for cheating (if I do get caught): \$1 million * 10% = -\$0.1 million Total expected Temptation Payoff: \$8.9 million Total expected Low Payoff: \$0.8 million Because \$8.9 million is greater than \$1 million, my incentive in Case I is to cheat. My incentive in Case II is also to cheat.

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1. 1. sdclark2 12:54 PM 4/3/08

I think this is a great statement of "the problem" at one level, but with teams such as say the NY Yankees where the owners directly profit from winning and even MLB profits from the "more exciting" behavior of doped up athletes, there is a lot of money going around that doesn't mind turning a blind eye. I doubt the commissioner is likely to make any serious changes without legislation and a tremendous amount of scrutiny and enforcement from outside the sports community.

2. 2. NorthWolfe 08:50 AM 5/13/08

Who is going to watch the Tour without the "super designer Athletes" being present?
Only people related to the sport, not the general public, judging from last years viewer fiasco.
With the massive public gone, with TV transmissions down to a minimum, with big sponsors gone, all your "High Payoff" strategies go down the proverbial drain. Sorry to bust up your party but people want to see records being broken, not a bunch of amateurs rolling back to 70's times. Same goes for the Olympics, if World Records don't fall audiences will. We all know what will happen with this scenario...

3. 3. Pontifex 08:52 PM 6/23/08

Dear Editor/Admin: Under the "Sucker Payoff" scenario in the Current Competition case, shouldn't the likelihood that a non-doping rider will get cut from a team for being noncompetitive be 90% instead of 50%? In other words, if the likelihood of my opponent getting caught cheating is 10%, then there is a 90% chance that he will not be caught cheating and, therefore, a 90% chance that I will be non-competitive (since I abide by the rules); which implies a 90% chance that I will be cut from the team. Another way to look at this is that the Expected Value of something is the probability of each possible outcome muliptlied by the payoff or loss associated with that outcome, and summed over all outcomes. This is basically a weighted average calculation where the relative weights are the probabilities, and therefore the sum of all the probabilities must add to 1. Notice how the sum of the probabilities in each of the other scanrios adds to 1 except for the sucker payoff. Would you agree?

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