Two otherwise equally competent athletes, Al and Bo, are engaged in a doping game. Each player needs to decide whether or not he wants to take the PED, hence there are two actions, "cheat" and "honest" available to them.

If they use the PED, the damage to their health is assessed to be 1.If they don't use the PED, obviously there is no damage to their health.

They are competing against each other in a tournament for a total prize of 4. If they both use the PED, neither has the advantage as it will cancel out; they will then split the prize at 2/2. If neither uses the PED, of course neither has the advantage either, hence they will also split the prize at 2/2.If one uses the PED while the other doesn't, then the player that uses the PED will win the whole prize of 4, leaving the other honest player with nothing.

The final payoff is calculated as the net gain, i.e. any potential gain from the competition (i.e. prize) minus the potential damage.

1. First, complete the payoff matrix (#1) based on the information provided so far. One cell has been filled out as reference. [Warning: it is extremely important that you get the payoff matrix right. It is the basis of all decision-making. Also, this is a large question, with many follow-up steps and revisions, so make sure to get this one right.]

Payoff Matrix #1:

		Bo
		Cheat	Honest
Al	Cheat
Honest		(2, 2)

1. Is there a collusive (or cooperative) outcome in this game? If so, please specify.
2. Is there a dominant strategy for either player? If so, please specify.
3. What will be the outcome of this game based on individual rationality? [In case that it isn't clear, please be advised that all gamesthis one and the next oneswill be based on individual rationality.] Please briefly explain your answer. Is this outcome a prisoners' dilemma?

So far, we have assumed that players who use the PED will never be found out later in their life (if they get to live another 7 decades, or 5 years) and hence there would be no damage to their reputation. With the technology advancement, that will change. We now assume that all players, at the time of deciding on using PED or not, know that eventually they would get exposed should they use the PED. As a result, while they get to keep the prize money (already spent), their reputation will be damaged and that will bring a payoff loss of 2. All the other conditions of this game will remain unchanged (for now).

1. Construct the new payoff matrix (#2).

Payoff Matrix #2:

		Bo
		Cheat	Honest
Al	Cheat
Honest

1. Based on this new payoff matrix, can you predict the outcome of the game (#2)?

We will now bring in yet one more new factor. Due to the Superstar Effect, the players who rise to the top of the game will potentially receive significantly higher prize money. The total prize money at stake is now 10 instead of 4, and it will be shared in a similar winner-take-all fashion. All other conditions of this game will remain unchanged, e.g. they will get exposed later if they use the PED.

1. With that, please construct the new payoff matrix (#3).

Payoff Matrix #3:

		Bo
		Cheat	Honest
Al	Cheat
Honest

1. predict the outcome of the game (#3).

1. (No more numerical calibration is needed.) Now, imagine that the international anti-doping agency will up the punishment very significantlyhow will that possibly change the game? Then, imagine the development of new and increasingly sophisticated PEDshow will that possibly change the game? Based on your findings of this entire game, please reflect on the difficulties in the campaigns of sports leagues and international anti-doping agencies, over the time, in keeping the games (sports games, not the game-theory games) clean and honest.