Recently my friend checked-in to a hotel and wound up in an interesting negotiation. He was staying for two nights and he wanted another room for the second night. He inquired if any rooms were available.

The hotel clerk explained that rooms were available, at a rate of $250. My friend countered that in his reservation, made weeks before, the rate was $150. Could he get the extra room for that rate?

The clerk explained he could not at the moment–they still had another day to sell the room for the peak rate. So my friend threw out an offer, “Well, let’s say I come back tomorrow evening at 6pm and the room is still available. Could you give it to me for $150?”

The clerk said my friend should check the next day with the manager.

Now ultimately he did not get the extra room, so in a way the negotiation was moot. But from an instructive perspective, the situation serves as an excellent case study in game theory.

There are two noteworthy details.

1. The negotiation over the room was like an ultimatum game. The hotel and my friend were negotiating over the potential surplus for the room. If they didn’t make an agreement in time, the room would go to waste–no surplus for either party.

2. The hotel was not willing to match the rate of the reservation, which presumably was a profitable rate. They risked the room being unoccupied. Why would they rather let a room go to waste than make some money?

This post is going to be about the second point, and the mathematics of auction theory can explain why a hotel might prefer to let its room go to waste on a given night.

Let’s explore why.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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Second-price auction

We will take a detour with a textbook example from auction theory. Imagine there are two bidders competing for a single item. The seller offers the item in the following auction. Each person submits a bid in a sealed envelope. The seller then compares the bids. The item goes to highest bidder who pays a price equal to the next highest bid. For this reason the auction is called a second-price auction.

While the auction might sound strange, it is actually similar to how many auctions work. For instance, think about the same auction held on eBay. The item starts at a low price and both people keep submitting bids. What point does the auction end? Well it ends when one person outbids the other. So the price of the item is one bid increment more than the lower bid, and the item ends up going to the person who is willing to submit the higher bid.

Or think about an open outcry auction, also known as an English auction. The auctioneer calls out prices that bidders agree to. When does this auction end? It ends when one person outbids everyone else–so the item sells for one bid increment more than the next highest bid, and the item goes to the person willing to pay more.

It’s a stunning result in auction theory that all of these auctions, and many other schemes, are ultimately variations on a theme, with the item efficiently going to the person willing to bid the most at a price of the next highest bidder. From the seller’s perspective, these auctions are all revenue equivalent in expectation.

So how much money will the auction generate?

The mathematics of even the simplest examples can be daunting. So I will cut to the results and interested readers can read the mathematical appendix.

Imagine there are two bidders who value the item between $0 and $100. Now the revenue for the auction will depend on luck. If the two bidders end up with high values, like $99 and $90, the item will sell for a high price of $90. But the bidders might also have low valuations like $20 and $10, so the item might sell for only $10.

In auction theory, we average out all these cases and calculate the expected revenue. The expected revenue turns out to be $33.33.

In other words, this auction will generate $33.33 in revenue.

Minimum price (aka reserve price)

Now we add another dimension to the auction. Suppose the seller incurs a cost to giving up the item. Let’s say the seller loses $10 when giving up the item. Now clearly the seller won’t just accept any bid. Obviously the seller will put a minimum price of $10 to at least cover costs.

The minimum cost is known as a reservation price. The second-price auction is modified as follows. If both bids are below the reserve price, then the good is simply not sold (“the reserve is not met”). If one bid is higher, but the other is lower, than the good is sold at the reservation price. And if both bids are higher than the reserve price, then the item sells for the second highest bid. (Another way to understand the rules: we can imagine the seller is a third bidder who always submits the reservation price as the bid. Then the auction is exactly a second-price auction with three bidders.)

The reservation price adds a twist to the auction. The higher the reservation price, the more the seller will get when the good is sold. But by the same token, the higher the reservation price, the higher the chance that the bids will be too low and the good will not sell at all.

So what’s the optimal reservation price? Again the math is complicated, so we will state the result and refer interested readers to the mathematical appendix.

It turns out in this case the optimal reservation price is $50. When the two bids are submitted, there are four equally likely scenarios.

–Both bids are too low, so the good is not sold and seller revenue is $0.

–Bid A is too low, but bid B is above the reserve. In this case, the good will be sold at the reservation price of $50.

–Bid A is above the reserve, but bid B is too low. Again, the good is sold at the reservation price of $50.

–Both bids exceed $50. The item is sold at the second highest bid, which is 1/3 of the way on this interval, at $66.67.

The expected revenue will be the average value of these cases, which turns out to be $41.67.

Look at that! Without the reserve, the seller was getting $33.34. Now, the seller gets $41.67, an increase of 25 percent in profit.

Comparing the two auctions, it’s clearly in the seller’s interest to have a reservation price.

The wasted hotel room

There is just one detail about the reservation price that deserves attention. It’s that in 25 percent of the cases the reserve is not met. In other words, 25 percent of the time the good is just not sold and goes to “waste.”

So the reservation price increases the payout to the seller at a potential cost to society if the reserve is artificial–there are times where a profitable trade could have been made but weren’t. By increasing the minimum price, the seller gets the higher valued customers to pay more, and that more than offsets the loss from lower valued customers.

The parallel to the hotel situation is pretty clear. A classy hotel wants to increase its occupancy rate, but it might be better off setting a high reservation price and letting some rooms go to waste.

Now clearly the hotel could earn even more by off-loading the unused rooms at discounted rates. And that’s the idea with websites like Hotwire or Priceline–the hotels offer the rooms confidentially. This avoids them having to discount rooms for higher valued customers while they also get more money from lower valued customers.

So my friend might have had better luck being anonymous rather than directly asking the clerk for a discounted rate.

Mathematical appendix

In the second-price auction, the price paid is equal to the second highest bid. So we first need to determine the optimal bidding strategy. This turns out to be very easy. Each bidder has a weakly dominant strategy to bid their valuation. This guarantees that a bidder (1) never wins if the price is higher than one’s valuation, (2) always wins if the price does not exceed the valuation–which is always profitable for the bidder.

Therefore, each bidder bids their valuation. Knowing this, one can compute the revenue generated–which will be the expected value of the second highest bid. This value will depend on the number of bidders and is equal to the 2nd order statistic of a distribution. In a uniform distribution from $0 to $100, the second order statistic for two bidders is $33.33.

The reservation price affects the auction as follows. Consider a reserve price r with valuations drawn uniformly from [0, 1]. The probability that both bids are too low is r2. There are two ways that exactly one bid is higher than the reserve, and so this happens with probability 2r(1 – r). The final case is when both bids are above the reserve, with probability (1 – r)2.

Now we write out the expected revenue for each case. When both bids are too low, the revenue is 0. When exactly one bid is above the reserve, the price paid is the reserve. When both bids are above the reserve, the expected revenue is the second-order statistic given that both bids are above the reserve. This is 1/3 of the way between the reserve and the value of 1, which is r + (1 – r)/3 = (1 + 2r)/3.

The expected revenue can be written as a function of the reserve, and we’ll maximize this function.

E(r) = r2($0) + 2r(1 – r)($r) + (1 – r)2($1 + $2r)/3 E(r) = 2r2 – 2r3 + (2/3)r3 – r2 + 1/3 E(r) = -(4/3)r3 + r2 + 1/3

Taking the derivative and setting it equal to zero, we find this function is maximized at r = 1/2 with a revenue of 5/12.

If we scale this to the example of $0 to $100, then the optimal reserve is $50 with revenue of $41.67.

This was just a small introduction to auction theory. For further reading, here is an excellent resource.

Chapter 9 of Networks, Crowds, and Markets: Reasoning about a Highly Connected World