Two Player Prediction Wagering

A bet is a tax on bullshit; and it is a just tax, tribute paid by the bullshitters to those with genuine knowledge.

– Alex Tabarrok

This year, twenty three billion (with a B) dollars was wagered on the Super Bowl. Almost all of this was through informal channels, such as friends at a kickback. I’d guess that many of these bets were informal “50-50” bets, the kind you used to make with your uncle. If your team wins, you get $10, if his team wins, he gets $10. For the more savvy participants, a Vegas money line might be googled to smooth out fairness in the odds. But what if Vegas hasn’t published a money line?

Prediction markets are an incredibly ingenious way for market participants to wager on niche scenarios that Vegas might not post a prop for. The highest grossing movie of 2024, if Taylor or Travis break up this year, or whether or not Trump says “skibidy” before the election are all events you could wager on in [1][2]. Unlike Vegas money lines, there isn’t a backroom setting a money line- it’s an order book representing what price individuals would trade the odds at. Because of this market behavior, there are many benefits offered by prediction markets that you might not see in Vegas; like the ability to sell your wager before the market closes.

However, prediction markets aren’t a miracle. The lifeblood of prediction markets is liquidity, and usability quickly evaporates as liquidity shrinks. Natrually, prediction markets will only host events which can attract betters, which by definition need to be public. How could you wager on a private event? What about one that doesn’t have obvious odds?

In the following post I propose a framework to conduct wagering between two parties on an event where a money line isn’t posted, and you couldn’t get enough liquidity to run a private order book.

Prediction Markets Basics

If you are familiar with prediction markets, or even basic EV concepts, you can probably skip this section.

A wager can be defined by two metrics: the cost, and the reward- what you pay to play, and what you take home if you win. Given these two values, you can calculate the implied probabilty of the event occuring. If your percieved probability is greater than the probabilty, you can accept, creating positive Expected Value (EV). In poker, “Pot Odds” would be a good example of weighing offered odds versus percieved odds.

Unlike poker, prediction markets normalize the payouts of a succesful wager. A won wager is worth one dollar ($1.00). Because of this nice, easy-to-work with number, the cost of the wager clearly indicates the implied probabilty. For a coin toss, an event in which exists 2 options that are equally likely, the cost of a wager on “heads” would be 50¢: fifty cents wagered, in an attempt to win 100¢. We know the 50 percent is the likelihood to hit heads. To guess the correct side of a 6 sided dice roll, a 1/6 chance, your cost would be 1/6 (.16), or 16¢.

Here’s the thing- we’ve been able to wager on coin flips for much longer than prediction markets have been around. What prediction markets offer is the ability to wager on the likelihood of events that have uncertainty surrounding them. Imagine the following questions:

Unlike the coin flip, none of these questions have gaurenteed probabilty. Similarly, the expected outcome to any of these can and will change as time goes on. If Kamala gets elected in 2024 [1], the chances of having a female president in 2028 goes up quite a lot. Events that impact her 2024 campaing would then impact the odds in question. If Kamala has a good polling week for the 2024 election, it also would raise her odds in the 2028 market.

Prediction markets allow participants to wager based off of any percieved information asymmetries- if you think you know something other people dont, they allow you to put money where your mouth is.

In practice, this manifests into a finantial free-for-all, attracting a mix of self-percieved intellectuals, degenerate gamblers, and all-around outsized risk-takers. This group tends to overlap quite a bit with people in the crypto world, and from from my observations leans to the right politically. There are some fascinating cultural dynamics in these communities, but that’s outside the scope of this post.

To recap: Prediction markets are cool, but they generally need 2 things for them to work:

  1. They need liquidity.
  2. To get liquidity, the events need to be happening in the public (politics, sports, pop culture, etc).

What happens if you don’t have that?

Private Events

let’s rewind to the inital question. how can 2 participants fairly wager on an event which doesn’t have a vegas money line, or isn’t on a prediction market? this could either be a fringe public event that wouldn’t have widespead interest:

will the golden state warriors start a player who originated from the g-league this season?

or it could be a private event (the more fun of the 2, and the inspiration for this thought experiment):

emma, danny, and i were set to meet at 3 o’clock. danny is not here, and it is currently 3:07. will danny show up by 3:15?

let’s review what we need to make a wager happen. we need an event:

danny will be here before 3:15

now we need our odds, as well as who is on either side of the action (buying vs selling). let’s start with the odds.

let’s say i believe there is a 40% chance of danny being here in the next 8 minutes. he’s often late, but not always super late. 40% is me thinking it’s certainly possible, but just a bit less likely than him showing up before. let’s call the 40% guess my “bid”.

Let’s rewind to the coinflip example, where we know the odds of the coin landing on heads is perfectly 50/50. Our EV is equal no matter which side of the bet we take- over many games, both Heads, Tails, and any mixed strategy between the two all end up breaking even. Now, let’s assume the odds of Danny showing up before 3:15 are exactly 40%, and both players know that. Despite winning less than half the time, your cost to buy would be lower than the cost of a 50/50 coinflip. Instead of paying 50¢ to play, you would pay 40¢, both to win the same dollar. Due to the less likely side being cheaper, and the more likely side being more expensive, either side of the wager should be equal value- meaning you could take either side, and be breaking even on EV.

Now, “assume the odds are 40%” is a pretty big ask- pretty close to complete bogus actually. The reality is nobody knows the odds, and that’s why we are betting in the first place. Let’s say that 40% is your best guess. You might think this is in the right ballpark, even though there’s some uncertainty. Emma is weighting the odds in the same way, but she has some information you dont; Emma met with Danny twice last week, and he was right on time both times. The odds in her mind are 80%.

Keeping Information Asymmetry

A wager is a claim to information. Your money is your wager, and your wager is only worth as much as the information you base it off. Giving up any of this info for free, especially in a zero sum game, would be naive and leaves room for exploitation. Let’s further our example and I’ll show how thats the case.

Let’s say Emma blurts out that she thinks there is a 80% chance Danny will get there by 3:15. If she truly believes this is the case, she would be happy to take either side of the wager, either buying or selling the action. Believing the odds are truly 40¢, you offer to sell her action at 80¢ a piece. Let’s simplify the outcomes to a binary: one of you is correct, and the other is not.

Option A: You are correct. The likelihood is truly 40%.

In this case, you are charging Emma 80¢ for something that should only cost 40¢. For each dollar that is wagered, you are making the difference in your price (80¢) vs what the action is truly worth (40¢), coming out to (80¢-40¢)=40¢ profit for each dollar you get down on the table. For a $20 wager, you are up 40¢ * 20, worth $8.

Option B: She is correct. The likelihood is truly 80%.

Like our previous coinflip example, both the buy and the sell are evenly correctly prices. No matter which side either of you take, both parties will break even.

If she’s right on the money, Emma will break even. If she is wrong, she loses big. In any case, revealing a bid first does nothing but harm. Between two keen wagerers, if an opening bid is required, no deal will ever be made.

Requiring an initial number to be thrown out also endangers the deal before it happens. Let’s say I hear 80%, and knowing this is much more likely than I predicted, assume Emma is working with better information that I am. This could dramatically impact how much money I put on the table, or might even cause me to walk away. While there are some interesting game theory concepts in someone leading with a bid that may or may not be a bluff, that’s not what I wanted to explore.

Sealed Bid Auctions

We can take inspiration from another competitive game, auctions. In particular, the Sealed Bid Auction. The execution for a sealed bid auction is simple: any number of participants privately write a single offer for the item in question, not knowing anyone else’s offer. After the auction concludes, the seller opens all the bids and accepts the highest.

Let’s steal this for our game. What if for our wager, both players keep their bids sealed? Emma and I secretly write a number down on paper, representing our bid from 0 to 100, representing our best guess at what the odds are. We both flip our papers, exposing both of our bids at the same time. This allows us to both get our bids on the table, without either of us giving up any opprotunity for exploitation.

Now we have 2 separate bids, but still no price. We also don’t know who will take the side of Danny arriving before 3:15 (buying the action for cost, at a chance to make $1), and who bets on him arrving later (selling the action for cost, at a risk of paying out the $1 prize).

Putting it all together

The cost \(C\) of the wager should be the average of your two bids. \({P_{a}}\) is Player 1’s bid, and \({P_{a}}\) is player #2’s bid, where each bid represent’s the players guess at the probability of \(X\) happening:

\[C = \frac{P_{a}(X) + P_{b}(X)}{2}\]

We now have a price for the wager. Let’s apply it to our made up scenario (I bid 40¢, Emma bids 80¢):

\[\begin{align*} C &= \frac{P_{a}(X) + P_{b}(X)}{2} \\ &= \frac{.40+.80}{2.00} \\ &= \frac{1.20}{2.00} \\ C &= 0.60 \end{align*}\]

Some fancy-looking math to show that we are splitting the difference. We now have a price, set at 60¢, but we still need to figure out who is buying and who is selling.

Well, if my percieved likelihood is 40%, paying 60¢ doesn’t make sense for me. However, if I can sell it for 60¢, I would be making a theoretical EV of 20¢ on the dollar. Emma believes the likelihood is 80¢, so she doesn’t want to sell it for less than it’s worth to her. But if she can buy it for 60¢, she believes she makes a profit of 20¢ on every dollar.

We can generalize this to say that the lower bid is the seller, and the higher bid is the buyer. Our bids are revealed, we know I am selling, Emma is buying, and we have our price. Who wins now?

Well, any wager like this comes with a great deal of variance. You might lose the bet even if you know Danny much better than Emma does. Even if you are getting money on the table at good odds, you will lose some wagers. The cool thing about this wager is that it yields positive EV for better predictions, at the cost of bad ones. If we model out a player’s EV playing this game, their profit (positive or negative) converges to the difference of their skill and their opponent’s over time. Two sharp players would break even over time, two fish will break even over time, but money will flow to whoever is smarter.

A bet is a tax on bullshit; and it is a just tax, tribute paid by the bullshitters to those with genuine knowledge.

It’s pretty common for people to think they are the ones with the knowledge, and significally less common for someone to acknowledge they are the bullshiter. If you have any desire to taxing the bullshit, start putting some money on it. Just beware sometimes you might be the one getting taxed. As the classic poker addage goes,

“If you look around the table and and can’t tell who the sucker is, it’s you.”

Footnotes

[1] Wrote the majority of this post before the election, but am publishing it afterwards.

[2] https://polymarket.com/event/will-trump-say-skibidi-before-the-election/will-trump-say-skibidi-before-the-election?tid=1731481733848