How much will you pay for a hundred-ish loonies?

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One of my favorite classes during my undergrad was Game Theory, where Prof. Joël Blit encouraged us to compete in games against our classmates. I want to share a game from that class that I particularly enjoyed.

In Canada, we call our $1 coins loonies.

A Canadian one-dollar coin showing a loon on the water.

I want you to imagine a jar filled to the brim with these loonies and a merchant selling that jar to anyone willing to bid for it in a crowded street. The value of the jar itself is negligible, and the value of the currency inside is what matters to you. You look carefully at the jar and inspect it from every angle you can, and you think that there are around

Your estimate 105 loonies

loonies

loonies in the jar. There are many other prospectors just like you, looking at the jar, trying to figure out how much value there is in there and how much they’d be willing to pay.

The merchant exclaims to the crowd, “Whoever places the highest bid shall win the jar and pay what they bid”. You think that there is $105.00 in the jar, so bidding more than that would mean you expect to overpay.

You decide to place a bid for

Your bid $104.00

Your estimate: $105.00.

Submit your bid, then read on to see what winning tells you.

Similarly, all the other prospectors around you are placing their bids. For now, assume they’re bidding close to their estimates of how much money is in the jar.

You may have won the jar, but do you think you are better off?

At this point, given that you won, you know two pieces of information. First, you believe that there is $105.00 in the jar; second, you placed the highest bid. If you only knew the first piece of information, you would’ve expected to make a profit of $105.00 - $104.00 = $1.00.

Suppose you placed the highest bid and the merchant says you won. Congrats!

But how should knowing that second piece of information change your expected profit?

Let’s take a step back and imagine the system as a whole. There is a true value of the number of loonies inside that jar. Let’s call it $v$ . Now everyone looking at that jar has their own estimate as to what $v$ is. Their estimate is $x_i$ , and if you collect all the estimates, you can imagine a distribution of estimates centered on $v$ , with their average $\bar{x}$ close to $v$ .

Example jar: 100 loonies

0 of 20 estimates. Add an estimate to see the average and highest estimate.

Example crowd’s estimates

No estimates yet.

As all the estimates accumulate, you notice that the highest estimate is generally well above $v$ . If everyone bids close to their estimate, winning is a reason to suspect that your estimate is too high.

The merchant determines that you placed the highest bid and says that you won. ~~Congrats!~~ I’m sorry.

Reflecting back on the situation we found ourselves in at the start of this game, we realize that winning the jar can be a curse, and we should account for that when deciding how much to bid, assuming that everyone else is bidding close to their estimate of how much money is truly in the jar.

Now let’s return to your actual bid and see whether you won.

Submit your estimate and bid above to reveal your auction.

if you want see what would happen in different situations:

How high does the highest estimate get?

Rival bidders20

No estimates yet.

Each dot shows the highest estimate from a completed crowd, in σ units.

Complete a crowd to add a dot.

Explore 1,000 simulated auctions

The actual jar value is fixed for each batch. These controls leave your original auction unchanged.

Rival biddersActual jar value ($)Average bidders' estimate ($)

Your bid

Simulated results from the same 1,000 auctions
Metric	Selected bid	Bidding the average estimate

Simulated average profit per auction ($), including auctions lost

Results vary with the sample. Lowering a bid does not guarantee a profit.

Now imagine bidding on a house. Buyers are trying to estimate its value, and winning could mean you were the most optimistic. As we saw, more bidders means more chances for someone to make a large overestimate. That gives you a reason to prefer a house with fewer bidders, especially if you’re buying it as an investment.

The same problem can happen in reverse. Imagine contractors bidding to renovate that house, with the lowest quote winning the job. Each contractor estimates how much the work will cost. Whoever underestimates the cost most could submit the lowest quote, win the contract, and discover that the job costs more than they’re being paid. The same thing happens to freelancers who win projects because they underestimate how much work it would take.

The interesting lesson here is that often you are placed in a situation where you expect to take an action based on your belief of the world. However, you may have a belief, and nonetheless, acting on that belief is a bad decision. You must account for the fact that you live in a society, and sometimes that means deviating from your view of the truth.

Thanks for playing!

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