Spice up dinner with pirate morality: a game theory problem to make paying the bill as entertaining as the meal itself
How to make your next dinner more interesting?
Another COVID weekend spent indoors as I wait for vaccine shots to release me from social isolation — sigh. What better way to use the time than dream about all the dinners with friends in the near future? As a conversational emcee of sorts, it’s time for me to start thinking of ways to make these dinners memorable. For this blog post, I’ve decided to make my next dinner spicier with some game theory flavoring.
I’ve recently been reading Israeli mathematician Haim Shapira’s game theory book Gladiators, Pirates and Games of Trust: How Game Theory, Strategy and Probability Rule Our Lives. Early in the book, Shapira discusses the decisions people make when ordering a group dinner. If the bill is being split evenly, should you order a more expensive dish knowing that everyone else will pay for it evenly? But, if everyone thinks similarly, will the whole group end up overspending, leaving everyone with a high bill and unhappy? If you plan on eating with these people again, what theory makes sense so that you’re not always picking up a disproportionate part of the tab or left ostracized as the person ordering Perking duck while everyone else has salad. In other words, how should you blend rationality with psychology? This problem is known as the Unscrupulous Diner’s Dilemma, and these are the types of questions the book is about.
Shapira also discusses the Pirate game problem, a well known mathematical game involving splitting up booty amongst a group of five self-interested pirates. The gist of the game is each pirate will propose how to split a 100 coin windfall, starting from the oldest pirate and continuing on to the newest. If the group agrees with a pirate’s proposed division, then the booty is split accordingly. If a proposal is rejected, the proposing pirate is forced to walk the plank and the remaining pirates continue until an agreement is reached.
Intuition suggests that the split will end with each pirate receiving 20 coins: a 20–20–20–20–20 split. This matches with our innate sense of fairness. The game theory solution is quite different. If all five pirates are rational, then the first pirate will allocate the plunder as 98–0–1–0–1. And a majority of pirates will vote to confirm this. If you don’t believe me, check out the math here.
Given I enjoy dining out in San Francisco, and I have nerdy and intellectual friends who will enjoy the chance to act as devious pirates, I thought I’d combine these two game theoretical games into a new scenario of my own creation: the Pirate’s Diner Dilemma.
The rules of the game
You and your friends are excited to see each other again. COVID has shuttered the restaurants in your neighborhood for too long. Now that you’re all vaccinated, you’re excited to get together after many long months spent isolated. But the conversation stalls — you’re all out of practice! As the bill comes, you decide to instill some fire in the group with devilish drama. Enter: game theory. Rather than divide the $100 bill evenly, you propose an alternative method:
- You (Archie) and your friends (Breanna, Curtis, Donna, and Edgar) will vote on proposals on how to divide the $100 bill.
- No player can be assigned more than $30 — nobody wants to be completely on the hook and leave feeling irate after all!
- You all are randomly assigned an order to propose a strategy for dividing the bill. For example, the first person makes a proposal. If it isn’t successful, the second person makes their proposal. And so on.
- If a majority of you and your friends agree to a proposed division, then the bill is split accordingly. If, however, a majority is not achieved (including a tie), then the person making the proposal is removed from the allocation discussion. They no longer make proposals or vote, but their friends can still stick them with the $30 max at their liberty. In fact, they might feel like the person deserves it for not proposing a better way to split the bill when it was their turn.
- You and your friends are pirate diners. That is to say, you act rationally and with self-interest.
- You’ve planned tequila shots at the next bar that will mend any hurt feelings, freeing you all to be your pirate selves. All diners therefore treat this bill-splitting problem as a “one-off” game.
Names are shuffled in a hat, and the order is randomly assigned. Assuming you and your friends are hyper-rational, what happens?
To your surprise and the amazement of your probability-minded friends, the order is randomly chosen and it’s alphabetical: you (Archie), Breanna, Curtis, Donna, and Edgar. You’re up and the whole table is looking at you. What should you do?
It is tempting to gravitate towards what feels like “logical” solutions. For example, wouldn’t a $20-$20-$20-$20-$20 split be well received? It might be boring — you were playing the game to not split the bill — but it seems fair.
Or couldn’t you propose a split where three diners pay $13.33 (one of them being yourself) and the remaining two pick up the max $30 bill? That would achieve the majority of three to two vote, since three voters would pay less than $20 under the even split scenario. Three people would be happy and only two friends would be out an extra $10.
But neither of these solutions is the outcome. Rather the final tally looks quite different. With a twinkle in your eye, you propose something that really gets your friends talking…
How will the other diners act?
Rather than start with the first proposal, it is useful to explore what happens if the first four proposals are turned down and only one diner remains: Edgar. In this situation, Edgar will propose a solution where the other 4 diners each pay the $100 (let’s say $25 each), while Edgar pays $0. Since he is the only diner left to vote, he will approve this strategy and leave the restaurant feeling quite clever. You and your other friends will leave with slightly lighter wallets, having paid for Edgar’s meal.
But Donna knows this is what Edgar will do if her proposal isn’t accepted. So she will select a strategy that is better for her and approved by both her and Edgar. Because she knows that Edgar gains by denying any proposal where he pays more than $0, she must also propose that he pays $0. Since the other players (Archie, Breanna, and Curtis) can no longer vote, she decides to assign herself $10, Edgar $0, and the other three diners $30 each. With this strategy, both she and Edger will vote to approve the proposal and leave the restaurant happy.
Curtis knows what Donna will propose if the proposals advance to her. So he must act accordingly. He knows a couple things: (1) Curtis needs 2 of the 3 diners to approve his proposal, (2) both Archie and Breanna can no longer vote, and (3) Donna will support a proposal where she pays less than the $10 she will be forced to propose for herself in the following round. Therefore, Curtis will propose that Archie, Breanna, and Edgar each pay $30. Of the remaining $10, he will pay $1 and Donna will pay $9, which is less than the $10 she’d pay in the next round. He and Donna vote to approve this proposal and leave the restaurant happy.
Breanna likewise understands what Curtis will propose next round assuming her proposal is rejected. She goes through a similar calculation as Curtis. She needs 3 of 4 people to support her vote, while you Archie can longer vote. So, she proposes the following: you will pay $30, Curtis pays $30, Edgar pays $29, Donna pays $8, and she pays $3. In this case, she will vote yes by comparing the proposal in this round to the one proposed in the next round. In this case, $3 < $30. Similarly, Donna ($8 < $9) and Edgar ($29 < $30) will also vote yes. You and Curtis stare on helplessly as you’re stuck with $30 bills each.
You know that Breanna will play rationally and divide the bill as listed above. With this information in mind, you need a strategy that will net you three of five votes compared to the alternative Breanna will propose if your proposal is denied. You need to make two people better off (in addition to yourself) and then stick two other people with the max $30 bill.
Since Curtis is already going to pay $30 next round, he’s your first target. You can improve his payout by offering him $1 better. After that you choose two diners to make better off while also maximizing your own benefit. That means you’ll slightly improve the two players already set to pay the most and then put the max check on the remaining people. You therefore decide to allocate $30 to Breanna and Donna. You’ve allocated $89 and decide to hit Edgar with the remaining $11 tab, leaving yourself nothing to pay. With this strategy you, Curtis, and Edgar will vote yes, while Breanna and Donna vote no. The game will end before advancing to the next round.
So here’s your proposal:
You offer to pay nothing! Your friends stare at you. But when the votes are counted, you leave without having placed anything in the collective plate. You are King Pirate.
This result feels wrong. It violates a sense of fairness: some people are paying more than others! Secondly, it defies logic: despite three people paying more than they would with a fair split, the proposal was still approved! This is reflective of the power of sequential decision-making and rationality in action. Since you made the first proposal (lucky for you!), you were able to set your strategy based on the decisions the other diners would make in response. Your goal as a pirate was not to maximize fairness, but to pay the least possible amount yourself. That’s why you won’t propose the $13.33-$13.33-$13.34-$30-$30 or $20-$20-$20-$20-$20 splits: you can do better.
The two people who end up paying the $30 max may grumble as they pull out their wallets — but they can take solace in knowing that their bill was the result of logic and chance (their randomly assigned proposal turn) rather than any ill-intent from you. Curtis feels bewildered and bemused as he ended up supporting an amount for himself higher than if you had split the bill evenly — he pays $29. You and Edgar whoop with hearty pirate laughter.
Hopefully game theory hasn’t ruined friendships.
Who would play this game?
I like this game because it highlights the tension between rationality and behavior quite clearly in game theory. Game theory assumes people will behave relationally rather than be influenced by emotions, psychology, cultural norms, etc. In fact, you and your friends may return to the even split. Why? You would presumably like to dine with your friends again and not leave on bad terms. Despite being rational actors, you will be emotional as well. And relationships will inevitably trump math. That’s why I included the rule that this was a one-off game with no hurt feelings to allow rationality to be the motivating factor.
So who would agree to play this game when splitting a check? Beyond straining relationships, there’s a 60% chance (three in five) that you end up paying more than the $20 if you did an even split. However, because the “winner” cases are much lower ($0 and $11), the expected payment actually remains at $20 mathematically: (20% x $0) + (20% x $11) + (20% x $29) + (20% x $30) + (20% x $30). Perhaps this can solve the diner dilemma of people over-ordering, knowing that they’re more likely than not to still pay more than an even split of the bill?
The $20 expected payment is also true of another check-splitting game that some (crazy) people play: credit card roulette. In this game (which is honestly more of a gamble), one person’s credit card is randomly chosen from a hat to foot the entire bill. The bills in this case are polarized, but the expected bill amount per person is the same: (20% x $0) + (20% x $0) + (20% x $0) + (20% x $0) + (20% x $100) = $20.
Given that friend groups play both credit card roulette and split checks, perhaps the Pirate’s Diner Dilemma could provide an in-between state? It’s more of a probabilistic gamble than an evenly split bill, but the winning and losing cases are less dramatic than credit card roulette. Since people tend to feel loses more acutely than equivalent gains (see: loss aversion), it could work. Pirate’s Diner Dilemma might not match the tense moment when a credit card is pulled from a hat, but it could provide several minutes of entertaining dialogue and debate as you and your friends strategize about what you’ll do and why.
Would you and your friends get to the game theoretical solution as hyper-rationalists or some other outcome?
Anyone played the Pirate’s Diner Dilemma?
If anyone tries this game with their friends, I’d love to hear. Additionally, you can play with more or less people or change the max cap based on your choosing to get other interesting results. Bon appétit!