A Game Theoretic Approach to The Ultimatum

Chad Broman¹, Tiffany Love²

¹ Department of Applied Psychological Machine Learning, Cranberry-Lemon University, Pittsburgh, PA, USA

² Department of Romantical Inference, Cranberry-Lemon University, Pittsburgh, PA, USA

Abstract

Chad has refused to propose to me so we’re going on this next season of The Ultimatum. In this game show we have encountered a new variant of the stable marriage problem in which we are presented with N heterosexual men and N heterosexual women who must match and have test marriages. I thought it was kinda hot there was a threesome show out there, but Tiffany says I’m not allowed to get physical during our trial marriages. Using the Gale-Shapely algorithm we can determine the most stable set of matching for our trial marriages in which Chad and I get with the better looking and less annoying contestants on the show. In this paper, we will show that by using the Reject-Accept proposal method of the Gale-Shapely algorithm with a mixture of gossip, we were able to achieve a set of matches where I got to spend my trial marriage with Todd and Chad got to spend time with Britney who I have an agreement with that there will be no funny business and Chad will be ready on the other side for Marriage. Also, Britney is pretty hot, so I won’t mind spending three weeks living with her. We confirmed that our method achieves Nash Equilibrium and is Pareto Optimized after we heavily adjust everyone else’s preferences to suit our interests.

Keywords: Game Theory, Gale-Shapely, Gossip, The Ultimatum, Pareto Efficiency, Nash Equilibrium, Competitive Dating, Reality Shows, Optimal Dating Theory, Trial Marriages, Failed Proposals

1. Introduction

We’ve basically been dating for over a decade, and Chad will not propose! I don’t want to give up our whole relationship, but I don’t know if I can keep dating him if we never get married. Plus, I love watching these reality dating shows it would be so much fun to be on one. I had to Ultimatum him.

I’m not ready to get married but I can see some benefits of going on the show. I didn’t like the idea of going on The Ultimatum at first, then I realized I could date someone else which could be fun. I realized I could test some of my relationship methodologies I’ve been using on Tiffany on someone else and double my sample size in only three weeks [1-4]. Because much of my research involves gaming and Ghost of Yotei just came out, it’s perfect timing!

By the end of this, I’m pretty sure Chad and I will get proposed to by our ideal matches Todd and Brittney. Also, I’ll have my answer regarding my romantic future. This could secure or blow up my future so I’ve got to make sure that I get Chad with Brittany who I think I can trust. Shannon could screw this up for us, but I’ll get into that in the results section.

2. Background

The Ultimatum is a game of love and strategy. If you love love like we do, it would be imperative to employ game theory to perfect our strategy for optimizing this trial marriage outcome. Assuming the contestants of this reality show are rational. I don’t see this blowing up in our faces. It’s important to understand some of the theory we will be employing. As has been mandated by Game Theory international law, we must explain everything in terms of the Prisoners Dilemma. To assist, here’s a graphic below to help out. Long story short, there are two prisoners, and both will benefit by ratting out the other but if both rat each other out, their reward is lower if no one rats anyone out.

2.1 Nash Equilibrium

A Nash Equilibrium is a condition in game theory where all players cannot improve their own condition by changing their own strategy. It was invented by John Nash a mathematician, US counterintelligence agent, and Emperor of Antarctica [5]. Chad, stop making fun of mental illness.

According to the prisoner’s dilemma, starting from any position, the double rat position where both people snitch is the Nash equilibrium because staying quiet as a strategy will never improve your own position. If no one snitches, you could commute your sentence by ratting out the other prisoner. If you’re ratted out, you will improve your own position by getting the other prisoner in trouble, cooperating with the guards, and forever putting into question the testimony of the other prisoner by spreading some pretty hateful rhetoric you’ll always be able to deny later with the other prisoners.

2.2 Pareto Optimization

Unlike the Nash equilibrium, a Pareto efficient condition is met when there are no options available that improves anyone’s situation without making another person worse off. In terms of the prisoner’s dilemma, all positions except for the double rat scenario, are Pareto optimum. No one can rat anyone out without decreasing the outcome for everyone else. It’s the nice-guy approach to game theory and the world would be pretty great if everyone used it. We’re going to try and make our solution pareto optimal, but there’s only so much you can do with these reality show contestants. Again…we’re assuming they’re going to be rational.

2.3 Gale Shapely Algorithm

Now let’s assume that the prisoners are divided into N tops and N bottoms and we have to move on to another metaphor Chad. Let’s not go there especially when the Gale-Shapely algorithm was made for a stable marriage problem and not for prisoner relationships which we assume are non-monogamous. Lets assume N heterosexual unwed men and N heterosexual unwed women in a village. The algorithm plays out like so:

1. All individuals rank members of the opposite set in order of preference
2. One of the two sets is chosen to make proposals
3. One individual from the proposing group in order of chutzpah who is not already engaged will propose to their most preferable option who has not already rejected them
4. Each person being proposed to will
   • Accept if it’s the first offer
   • Reject if this is worse than their current offer
   • Accept if this is better than their current offer. In this case they will jilt their previous offer
5. Repeat steps 3 and 4 until everyone is matched

Once everyone is engaged, this algorithm will create a set of stable matches in which no proposer or proposee will desire someone else who prefers them over their current match. It’s the algorithmic way to get everyone to settle down and get off those apps. Assuming the guys will all make the first move we may apply this to the initial stage of The Ultimatum.

3. Ultimatum Roster 

In order to set up the parameters and constraints of our application of the Gale-Shapely algorithm, we must understand the starting condition of couples, and everyone’s preferences. We’ll also discuss the deal breakers and who gave each ultimatum.

3.1 Couples

This season is beginning with a total of six couples which can be seen in table 1. It will be important to note who issued the ultimatum as many participants who issued an ultimatum will also want a fellow ultimatum giver and vis versa. This is true for the women but not the men likely because the only woman to receive an ultimatum was Zoe and she’s kind of nuts. So far, she’s already started four fights on set. She is a little hot even according to the hot-crazy spectrum [6]. But Chad, women like that never get matches like the ones on this show! Also, she’s just been causing drama with the other women like when she always says my clothes are a nice Good Will find in front of everyone! You just don’t like her because she’s been flirting with me. Well, she flirts with Marcus more so I wouldn’t get any ideas Chad.

Man	Woman
Chad	Tiffany*
Todd*	Zoe
Evan	Shannon*
Marcus	Brittany*
Liam	Harper*
Diego	Celeste*

3.2 Men’s Preferences

As you can see from table 2, the male preferences in this game are all appearance based. They won’t let me take pictures so this may need to be proven in follow on research or when the show finally airs. The producers were trying to draw in a more visually driven crowd for this season. Hey, the guys are also hot, and we have all had some really good conversations around the hotel pool about commitment, love, and stuff. Why do all the conversations have to happen around the pool? It speeds up the dating process, okay. Stop trying to ruin a good show just when I started to get into it okay?

Man	Preference List
Chad	Brittany, Zoe, Celeste, Harper, Shannon
Todd	Brittany, Shannon, Celeste, Harper, Tiffany
Evan	Harper, Zoe, Brittany, Celeste, Tiffany
Marcus	Celeste, Zoe, Tiffany, Shannon, Harper
Liam	Shannon, Brittany, Celeste, Tiffany, Zoe
Diego	Harper, Tiffany, Shannon, Brittany, Zoe

3.3 Women’s Preferences

As previously mentioned, Todd jumps to the tops of everyone’s list as the one male ultimatum giver. Marcus and Brittany’s situation is complicated. It involves some weird connection around some pretty deep sci-fi fandom connection that’s been off putting to the rest of us. Brittany is another exception because I have talked her into handling Chad’s trial marriage. There are a few things he’s got to learn the hard way and I think she’s the ticket. Shannon is also interested in Chad and Todd so that may be a problem for me later.

Woman	Preference List
Tiffany	Todd, Marcus, Diego, Evan, Liam
Shannon	Chad, Todd, Liam, Diego, Marcus
Brittany	Chad, Todd, Evan, Liam, Diego
Zoe	Marcus, Liam, Diego, Evan, Chad
Harper	Todd, Evan, Diego, Marcus, Chad
Celeste	Todd, Liam, Marcus, Evan, Chad

Given the strong preferences towards Chad and Todd, I have some work to do to get with Todd for my trial marriage, especially if we are using a Gale-Shapley algorithm. Todd’s a sensitive soul and I think if he gets a chance to trial marriage with any of these girls, they are not getting back with their current partners after the trial marriage. I’m really going to manipulate everyone’s preference lists for their own good. They don’t deserve him like I do. Also, you’re at the bottom of his list so there’s a slim chance he takes you away from me.

4. Strategy

There’s nothing in the Gale-Shapley algorithm that dictates that everyone’s preferences have to stay the same. Additionally, the order of proposing is equally important. Suppose one of the proposers is publicly and humiliatingly rejected such that he drops far down everyone’s list and does not have the confidence to ask anyone else out until everyone else is coupled up. It only has to be embarrassing for a few days to control everyone’s preferences otherwise it will not be stable.

I have convinced Brittany, or we have convinced Brittany that Chad will be the perfect trial marriage. We had a really great convo; she’s on the hook for me. I’ve been staying regular on my back and bi days so I know why. Also, she felt really bad after I told her about the whole worm thing. Okay, just don’t go telling any of the other guys about that. I’m pretty sure that’s why no one’s interested in me! Why do you always have to trash talk me in public about how I acted like a worm for that one experiment [7]. That’s also part of my plan; our plan. Todd will propose to Brittany first, and she will accept according to the Gale-Shapely algorithm. Then Chad proposes to Brittany which will then accordingly have her embarrassingly publicly jilt Todd. She told me she’s gonna do it in front of everyone.

Todd will be so shocked that he was jilted that it will be a while before he feels comfortable proposing to someone else. By that time, everyone else will already have made their proposals and the gossiping will allow Todd to drop in everyone else’s preferences. Brittany will be with Chad and that’s a lock because we arranged it. Evan will propose to Harper which is her second choice. Marcus will propose to Celeste, then Liam to Shannon, then Diego to Harper who will reject him, so he’ll propose to me. I’ll accept, but only because he’s the first to propose to me.

When Todd comes to, he’ll feel too awkward and skip Shannon, fingers crossed, who has been a little intimidating and start with Celeste. She’ll be comfortable with her current match Liam, originally her second choice, and know how much Harper wanted to get with Todd. Also, his rejection will be extremely embarrassing. The same will occur with Harper who has her second choice. Then he’ll come to me, I’ll proudly Jilt Diego who will then get rejected by everyone except for Zoe who has yet to be paired.

5. Implementation

It all went perfectly with one possible complication; I may have flirted a little too much with Shannon, but she was really asking for it by the way she was looking at me. Shannon wanted to get with Todd and then me, and she almost blew the whole plan. She felt so bad for Todd after Brittany jilted him and then was jealous that I proposed to Brittany instead of her. Our plan’s primary flaw was that Todd would feel too awkward to get with Shannon. When he proposed to Celeste and was rejected, she felt so bad that she almost reached out.

That’s when I came in with some mojito’s which we’re a little stronger than usual and Shannon was too distracted and tipsy to reach out to Todd before he was rejected by Harper and accepted by me the last night of filming before the trial marriages had to begin according to the show filming timeline. You don’t sleep on love. I certainly didn’t and was not passed out in my hotel room when Todd had the courage to ask me to be his trial wife!

6. Results

So, we got with who we wanted and as it stands, we’re in a Nash equilibrium and Pareto Optimization. No one can take my Todd away and keep Pareto optimality because it would hurt my feelings badly. The resulting couples can be seen in table 4.

Man	Woman
Chad	Brittany
Todd	Tiffany
Evan	Harper
Marcus	Celeste
Liam	Shannon
Diego	Zoe

Thankfully, Liam never talks to Todd, otherwise this whole thing could blow up in our faces. At least, when they do talk it’s about the Seattle Seahawks, since they’re both football dorks and never get around to talking about relationships. You can’t blame them, they are having a great season this year. Okay Chad, you can’t be into video games and football so you have to pick one. Anyway, without talking to Liam, Todd will never know he could have gotten with Shannon. I never expected her to still consider him after that embarrassing rejection me and Brittney coordinated.

Low key, Todd and Shannon do not like me. Todd hates that I’m with Brittney now and Shannon hates that I’ll be with Tiffany at the end of this and not with her. Why don’t we call it jealousy. Thankfully, I’m so happy with ending up with Todd that there are no possible moves someone could make that wouldn’t hurt someone else. If everyone’s cool with not ruining my day and achieving Pareto efficiency, I think we need to keep the matches where they are at.

It may not look it, but this list of final couples is at Nash equilibrium. According to the original preferences, the current state of our matches would not be in a Nash equilibrium because many of the girls would have rather gotten with Todd than their current matches. This is only true outside the limited confines of The Ultimatum. I’ve had a lot of conversations with all the women on this show, and I can say that they may have been happier during the trial marriage, but I don’t think they could hold that man down. Todd is like heroin. A taste is ecstasy, but a full meal could ruin your entire life. But he won’t ruin your life, right? Tiffany?

6. Conclusion 

We have found Pareto and Nash Equilibrium using the Gale-Shapely algorithm in this upcoming season of The Ultimatum. There are no moves left that won’t hurt someone else which achieves Pareto efficiency, and there is no strategy that we may all adopt that improves anyone else’s outcome achieving Nash Equilibrium given that only I can handle Todd; or I will destroy them. It’s a show lighten up. We look forward to the relationship lessons and fun times we will have during the trial marriages of this reality show and will report on our findings in future papers.

References

1. Chad Broman 2021 A Time-Series Analysis of My Girlfriend’s Mood Swings :: How to Prove Anything: 30 absurd research papers no one else was brave enough to publish
2. Chad Broman 2022 Sub-Nyquist Sampling While Listening to My Girlfriend :: How to Prove Anything: 30 absurd research papers no one else was brave enough to publish
3. Chad Broman 2016 A Conversational Bottleneck: Talking to Tiffany While Playing Dark Souls :: Journal of Psychological Machine Learning
4. Chad Broman 2019 Tiffany Nyquist Sampling Frequency: A Novel Approach to Beating the Sekiro Guardian Ape :: Journal of Psychological Machine Learning
5. John Nash Wikipedia page https://en.wikipedia.org/wiki/John_Forbes_Nash_Jr.
6. Chad Broman 2015 The Hot-Crazy Spectrum: A Time Tested Methodology for Optimizing Partner Selection for Romantic Enjoyment :: Journal of Bros on the Town
7. Tiffany Love 2024 Would He Still Love Me as a Worm: Indirect Sampling and Infereence Techniques for Romantic Assurance :: How to Prove Anything: 30 absurd research papers no one else was brave enough to publish

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