Lottery Problem Statement
Imagine there is a promotion company offering 1 million people an opportunity to enter one of two unique lotteries for free. You can choose to receive one ticket to enter either a $10,000 drawing or a $2,500 drawing, everybody is given the same choices. Once you made the decision, you will enter the corresponding drawing. There will be only one winner in each lottery, the winner will take the entire prize pool ($10,000 or $2,500). So how would you make the decision?
The problem might seem simple at first glance, but the situation is rather dynamic because a single participant’s optimal strategy can be affected by other participants’ decisions. Let me explain why.
Let’s say for some reason you are given the opportunity to enter the lottery after everybody else have made their decisions, that means you gain an unfair advantage by having knowledge on participants’ distribution over the two lotteries. Say x participants decided to enter $10,000 drawing, while (1 million -x) entered the $2,500 one, we can calculate the expected return based on this information.
Expected return on $10,000 drawing: 1 in x probability to win $10,000 = 10000/x
Expected return on $2,500 drawing: 1 in (1 million – x) probability to win 2,500 dollars = 2500 / （1 million –x)
Given any value of x we just to need to calculate which expected return is greater and proceed to enter that particular lottery. Sounds simple right? but in reality you will not be able to gain this type of unfair advantage, and in most cases you will not know what other players’ strategy. So how would we go about making an optimal strategy without knowing other players strategies?
Game Theory and Nash Equilibrium
Game theory is the study of strategic decision making. More formally, it is “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.”
The lottery problem is a perfect example of how to use Game Theory and mathematical model to address optimal strategies among multiple competing decision making processes. Of course, poker is another perfect example which we’ll cover in our next blog post.
In game theory, the Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally.If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.
In simpler words, deriving a solution from Nash Equilibrium will give you an un-exploitable strategy that nobody else can beat even if they know exactly what your strategy is!
Since we can make an safe assumption that nobody knows what strategy others employ. The un-exploitable solution strategy derived from Nash Equilibrium also becomes the optimal strategy. Now let’s try to find the solution together.
In order to solve this problem with mathematical models, we need to make couple necessary (potentially questionable) assumptions:
- Every player given the opportunity will be participating,
- Every player is rational – they will try maximize their expected money return,
- Every player has linear utility – the amount of money they receive is proportional to their happiness.
There are two pure strategies in this game.
- Enter the 2,500 drawing,
- Enter the 10,000 drawing
There are infinite family of mixed strategies: enter the 2,500 drawing with probability p, the $10,000 drawing with probability (1-p).
If everybody is rational, they will come up with an optimal strategy based on public information, which is a mixed strategy (p, 1-p). This would give us:
10,000 Drawing: 1,000,000 * p participants yields profit of $10,000 / 1,000,000p
2,500 Drawing: 1,000,000 * (1 – p) participants yields profit of $2,500 / 1,000,000(1-p)
Let’s pull in the Nash Equilibrium concept now, if everybody is employing the same optimal strategy, how do we make sure it’s indeed optimal? A player of an optimal strategy must not have an incentive to switch to a different strategy. This happens only when
solving above equation gives us p=0.8, so each player should choose the $10000 drawing with probability of 0.8 and/or the the $2500 drawing with probability of 0.2.
Make sense? Still in doubt? Let’s say I will use this strategy play the same lottery game with you. Will you be able to find a better strategy that’s statistically optimal in the long run?
In my next post I’ll continue discussing about Game Theory, we’ll also shift our focus to a more strategically oriented game called No Limit Texas Hold’em.