A game theoretic approach to credit card roulette

Factual Questions
1

A new friend of mine this evening introduced me to the concept of credit card roulette. Credit card roulette is when a group of people go out to dinner and decide to have the waiter blindly pick one credit card from the pile at the end and charge the entire bill to it.

Now, from a game theoretic perspective, it’s obvious that such a game introduces a prisoner’s dilemma problem. Assuming everyone knows about the game before they make their menu choices, each person individually will be incentived to pick something more expensive than what they would normally get. And yet if everyone picked something more expensive, everyone ends up paying more than what they would have individually.

Now making the simplifying assumption that each person can only choose one item on the menu and utility is monotonically increasing with price, then the Nash equilibrium is obviously everyone choose the most expensive thing on the menu. However, if we assume that this is a regular dining group that eats out every week, then there seems to be some sort of equivalent strategy to tit-for-tat which can do better than the Nash equilibrium but I can’t seem to figure out what that is.

Now, it seems like the best result possible from the menu ordering game should be the equivalent of how people would order if they were paying for their own bills. However, assume that everyone at the table is also a gambling fiend and they derive some utility g from playing credit card roulette. What is the optimum strategy and how large would g have to be in order for credit card roulette to be preferable over straight ordering?

If it helps, you can make the additional simplifying assumption that menu prices are a continuous curve rather than discrete choices.

During dinner, I came up with a potentially interesting strategy which is that everyone writes down their menu choice and it all goes in a big pile and people pick a dish randomly. Then, people are allowed to negotiate trades in dishes until everyone is happy with their choice. It only occurred to me after I got home that of course this couldn’t be better than nash equilibrium because there’s no persistance of information across games. Still racking my brains about how to apply something like tit-for-tat.

Edit: Playing credit card roulette at a bar leads to a situation in which the optimal strategy is far from optimal in so many different ways. :frowning:

2

Let’s assume the most expensive dish available costs $50. If there are four people involved in your game, and everyone picks the $50 dish, then the total bill will be $200 (let’s ignore, for simplicity’s sake, tips and drinks). The estimated cost, in the sense of the vaue of the game, of the dish for you would be $50, because there’s a 25 % chance of you ending up to pay the $200.

Suppose, now, that you, and only you, switch and pick a cheaper dish at $25, which is also the dish you would chose if everyone paid for his or her own bill separately. The total cost of all four players will be $175, and the value of the game for you would be $175/4 = $43.75. IOW, switching to the cheaper dish will decrease the cost of the evening for you by $6.25. It decreases these costs by only a quarter of the savings the switch will produce for the total bill, but there still are significant savings for you to be gained from switching.

I see that you assume the utility of the menu to increase proportionally to the price of the dish. In this case, the increase in utility from ordering the $50 dish will outweigh the increase of costs, making the choice of the expensive dish preferable, as you point out. But the assumption that utilty increases proportionally to the price of the dish makes all options equal if everyone follows this strategy. If all the players take the $50 dish, then the estimated cost of the evening for every player will be $50 as compared to $25 if everyone were paying separately. But since the utility of the dish decreased by the same rate as the estimated costs, none of these two scenarios is preferable over the other.

3

I’d like to add that I want to enjoy my meal and the company, and if, for example, I got stuck with the whole bill twice running, I wouldn’t enjoy myself.

4

I said it was monotonically increasing. Proportionality would indeed make the two games equivalent. Most likely, your marginal utility for every dollar increase is decreasing.

5

From a strictly game-theoretic POV, Schnitte nailed it in one, net of your comment just above. The shape of MY monotonically increasing utility function versus MY expected value determines MY optimal strategy. Ditto for each other player. The problem is MY expected value depends on the sum of everybody ELSE’S monotonically increasing utility function, which I cannot know.

And without knowing all those, any more specific answer can’t be had.

If you want to make the simplifying assumption that everyone has the same function regardless of its specific details, I suspect (but cannot prove for lack of skill) the answer will reduce to the same as for the proportional case.

So much for theory, on to practice …
The real-world answer lies within your statement that these are “gambling fiends”.

The essence of human intuitive gambling behavior is to overestimate the odds of successful outcomes, while underestimating the odds of unsuccessful ones. In addition, peoples’ utility function for gambling itself is also lopsided in that a win feels more good than an equally-sized loss feels bad.

These two behaviors self-reinforce, which is why somebody will feed their week’s paycheck into a slot machine in exchange for three $10 “jackpots” along the way.

Applied to the problem at hand, the players should choose the most expensive item to maximize their psychological utility, if not their budgetary utility.

6

Having nothing to do right now, I’ll take a closer look at it assuming a non-proportional but logarithmic utility function. I don’t really have a goal which I want to reach with these calculations, I just calculate along to see what I get.

Defining u as utility and d the price of the selected dish in dollars, let u = ln d.

If everyone pays separately, and everyone has the same utility function stated above, then choosing the $50 dish will yield a utility of 3.91, or 0.078 per dollar, while the $25 dish will yield a utility of 3.22, or 0.129 per dollar. Thus, if you have a fixed budget available every month for dining in restaurants, you should take the $25 dish.

Now let’s examine your available choices in a party of four persons dining together under the rules of credit card roulette.

If everyone else takes the $25 dish and you do the same, then the average cost of the evening for you will be $100/4 = $25 for a utility of 3.22, or 0.1288 per dollar. The same result as if you were dining alone.
If everyone else takes the $25 dish and you take the one at $50, then your average cost will be $125/4 = $31.25 and the utility will be 3.91. You get a utility of 0.1251 per dollar. That’s less than you’d get by taking the $25 dish, so you should take the cheaper meal.

If everyone else takes the $50 dish and you do the same, then the average cost of the evening for you will be $200/4 = 50 for a utility of 3.91, or 0.078 per dollar. Same result as if your were dining alone.
If everyone else takes the $50 dish and you take the one at $25, then your average cost will be $175/4 = $43.75 for a utility of 3.22, or 0.0736 per dollar. That’s less than the utility per dollar for taking the $50 dish in these circumstances.

In other words, there is no dominant strategy here for you to play. If the other players take the more expensive dish, you should do the same; if the other players take the cheaper dish, you should do the same. I guess this means there is no prisoner’s dilemma in this constellation - the prisoner’s dilemma, IIRC, requires that every player has a dominant strategy which is thus followed by all players, resulting in a lower payoff than they would get if everyone chose a non-dominant strategy.

I realize this is a result of the assumptions from which I departed. It should be possible to construct an example (different dish prices, different utility functions) in which you’d end up in the classic prisoner’s dilemma. Goes to show that there cannot be a definite solution to the problem unless the OP gives more information.

To generalize a bit, I think it should be possible to plot the utility functions for any possible choice for any player, and the average cost of the night function for any player, and based on this we could get the utility per dollar function for any player. We could then determine the maximum of the latter function and see if there’s a dominant strategy. Then we could see if the application of the dominant strategy by all the players results in a lower payoff than possible for another combination of strategies, in which case we have a prisoner’s dilemma. The way to evade this dilemma is the same as in other instances of it: The players should (either explicitly or implicitly) agree not to take the dish which is their dominant strategy. If the game is played repeatedly, then there will be a strategy akin to tit for tat which punishes deviations from this agreement. How this tit for tat would look like depends on the exact numbers and utility functions.

7

Oh, and although you’re very likely to be aware of it, I’d just like to point out that this game is mathematically identical to the dinner in which the total sum is divided evenly among all the patrons. Has any research been done on this - definitely more common - situation?

8

As has been pointed out, this problem is like the Prisoners’ Dilemma. The best outcome is everyone cooperating (ordering the cheap meal), but each person (with no outside enforcement) individually has an incentive to not cooperate. This leads to the worst outcome.

Also as was mentioned, one way to have a subtle kind of enforcement is by playing the game repeatedly. The repeated Prisoners’ dilemma has been the subject of more than one tournament. People submitted computer programs that played against each other repeatedly. The finding was that strategies resembling tit-for-tat did the best in the long run. That is, strategies that cooperated until someone else defected and then punished by not cooperating next time. It was important however, to not punish too much. See

for more details.

9

There’s a decent article here.

10

Interesting, but I think you need more data points. Many of us have credit cards which accumulate airline miles or membership reward points. This would have the effect of decreasing the penalty to the one who pays, and I think should be factored in to the equation.

How to do this, I will not even presume to guess. Just consider it a :slight_smile:

11

Is this strictly a mathematical exercise, or do you plan to apply this in the real world? I ask because there’s a huge variable that’s being overlooked. You’re automatically assuming that the more expensive dishes provide more enjoyment, unless you say that your only enjoyment is gaming the system.

12

No effect. The net effect of incentive programs like that is to effectively decrease the cost of everything by some fixed proportion. So instead of looking at a $50 dish and a $25 dish, we’re looking at a $49 dish and a $24.50 dish. The analysis is the same.