On expected values and game strategies

See, I haven't done a truly nerdy/academic posting in a while, so I figure it's time, especially since my mother has taken up Sudoku (has anyone not at this point?).

Speaking of whom, her same bizarre self (who insisted I teach her how to play Texas Hold 'em over the holidays...I have this terrible image of her exclaiming "Read 'em and weep boys! Woo hooo! Kiss the pretty ladies...oh yeah..." *shudder*) introduced me to "Deal or No Deal," a deceptively simple little game show.

Be afraid: it's a game show on CNBC.

Anyway, the game is simple: you, as a contestant, are presented with 26 suitcases guarded by beautiful models (too thin for me, but that's another rant). Not really guarded, mind you...you don't get attacked by a frothing but beautiful woman if you get too close...that would be cool if unsanitary...rather, guarded in the Vanna White sense of standing near for no apparent reason. Each suitcase contains a dollar amount, generally between one cent and one million dollars, and these amounts are all listed on a board. The amounts are at standard, currency-like denominations. Unfortunately, the cash itself isn't in the case, which is something I think they should do...at the moment it's just sparkly signs like on that plebian game show The Price is Right. Pshaw. We cannot be bothered to consort with the peasants. You! Fetch me the Wall Street Journal! You! And you! Fight to the death!

Where was I? Oh yeah. So you pick a suitcase but don't open it. That's your suitecase, and you get the amount of money listed in it unless during the course of the game you do something to fuck it up. Now the game begins. You start by picking 6 cases to open. The pretty ladies do so, and 6 values get removed from the board. Then the "banker" calls and offers you some amount of money based on what's left on the board. If you take it, you get the cash you were offered, and the game stops. Otherwise, you keep going, open another 5 cases, and repeat.

This cycle continues, with the number of cases you have to open in the intervening period decreasing each time until you are only opening one at a time, until you stop the game by taking the deal or until you reject the deal offered to you with two values left on the board, in which case you open your case and get whatever value is in there.

What value gets offered by the banker, you may ask? Well, they don't tell you. In game terms, it's the value the banker is willing to pay to stop you from going forward and potentially extracting more money out of him. More mathematically, it seems to be some percentage of the average amount left on the board (I'm thinking about 60% of the average of what's on the board). Probably the formula is a little more complicated than that, but it's a good estimate.

So I spent waaay waaay too much time last night, while I couldn't sleep, trying to figure out what the ideal strategy for this game is. I quickly realized there is a problem.

In game theoretic terms, the optimal strategy is to never accept the deal and to always take what's in your case. Here's why: at any given stage of the game, the expected value of what's in your case is going to be the average of the values left on the board, yet we just said you will be offered some percentage of the average of what's left on the board, ergo, you should always reject the deal. Ergo, if you always reject the deal, you'll eventually open your briefcase.

Here's the rub though: the expected value, especially of a game like this, tells you only in aggregate what the payoff of a game will be. That is, if a million people play the game, the average payoff of all of them will be the expected value of the game. Great! Except for one thing. You're not a million people. You're one person, and you get one chance to play the game. For you, an offer of $200k might be worth more than a 1/3 chance of $1 million and a 2/3 chance of $0.

So, I haven't quite figured out how to reconcile that. I haven't figured out how to reason about the ideal strategy for a single instance of a game. I have a feeling there will be scads of information about this in the options trading literature and I'll have a "duh" moment when I find it, but for now I'm left scratching my head. I could also ask the lab's game theorist, but I think he'll tell me I'm stupid. That seems to happen a lot.

Incidentally, my pseudo-strategy is to reject the deal so long as it's likely (i.e., >50% probability) that you can make the average value of the board go up in some subsequent round. Specifically, if it's likely that there will be a tile greater than or equal to $100k on the board when you're done choosing, keep going. Try it yourself.

Interesting experiment for the procrastinating: play some relatively large number of games (greater than 26...like 30 or 50...or more!) always declining the deal. Keep track of your aggregate score. Then do the same thing with my strategy (or your own) and see how you do. My guess would be that your average is going to be higher in the playing-to-the-end strategy, but other metrics like your minimum payout, standard deviation of your scores, number of payouts less than, say, $50k, etc. will likely be better with my strategy.

I should probably go to bed now, huh?