Positive Expected Value Lottery – We’re All Gonna Be Rich – A Game Theory Problem

In case you haven’t heard, the MegaMillions jackpot is up to $500 million.  Yes – half a billion dollars.  A monstrous sum, to be sure.

Expected Value hounds like me immediately think: “+EV!” – if the odds of winning the jackpot are roughly 1 in 175mm, then a $500 MM prize gives each ticket a positive expected value.  Now, of course, there are other things to consider:  1) you have to pay taxes.  2) if you want the lump sum payout, you get much less than $500 MM (about $359 MM according to their website).   3) there are secondary prizes to account for.    Most importantly, and here’s where the trading analogy comes in:  you have to consider proration risk – the risk that someone else will have the same winning ticket and you’ll have to split the prize.

Now, you can work out some fun math on this – but it’s kinda a moot point for the purposes of this post,  because you can’t actually physically buy all 175mmm combinations of numbers to guarantee that you’ll win.

But what if the Lottery offered you a structured product:  the “All The Numbers In One Ticket” product that would guarantee that you would split the jackpot with all other winners?    This is a game theory question without an easy answer.   In trading scenarios, like the Norilsk Nickel tender, we usually know our worst-case proation risk, and we can estimate the post-tender value of the shares.  The problem with my theoretical pseudo-lottery is that we can’t really estimate our proration risk*.

My Game Theory Question of the Day, simplified, is this:   Assume that we have a lottery where the winner gets $500 MM in cash, tax free.   Assume there are 175MM possible number combinations, and that it costs $1 to play each combo.   How much would you pay for a product that offered you every combination of numbers – something you might like to purchase, based on your expected value calculations?   You will have to split the jackpot with everyone else who buys this “Guaranteed Winner” ticket, as well as with any “normal” person who just happens to hold the winning combination.  Ignore taxes.

What is the “fair market price” of this “Guaranteed Share of the Winnings” ticket?

The question is somewhat confusing, so let me elaborate a bit by example:   If this ticket cost $1000, you’d expect lots of people to buy it.   The Lottery will sell it to anyone who wants to buy it at the price determined by the market.     If this ticket cost $200 MM**, you wouldn’t expect too many people to buy it, but you know it’s still positive expected value if there’s only one winner (spend $200 MM, win $500 MM) – so it might be rational to expect more than one person to buy the ticket…   But if three people each pay $200 MM for the special magic ticket, it’s no longer +EV…   A prisoner’s dilemma, it seems.

I don’t think that there’s a quantitative answer to my question, but feel free to expound in the comments.

-KD

* in the real world, in fact, we can estimate the proration risk – we can estimate the expected number of jackpot winners based on the number of tickets purchased.   My thought exercise here  is one of  Game Theory, not practice, though.

**  the price may very well be more than $175 MM because you can’t physically buy all 175MM tickets – you’re paying for convenience.  In reality, my guess is that the real price would be lower, to account for un-quantifiable proration risk.

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