# Playing with models: quantitative exploration of life.

## The optimal strategy for setting the price of event tickets

Posted in Economics by Alexander Lobkovsky Meitiv on September 10, 2015

I love tennis and always visit the ATP tournament in DC in August. This year I went with a friend who bought the ticket at the last moment online and discovered that it was deeply discounted (almost 50%). That fact coupled with the observation that the stadium as less than half full got me thinking: are the organizers employing the best ticket price strategy to maximize the receipts?

To determine the optimal ticket pricing strategy we need to know something about how people make buying decisions.  There are a number of different ways of describing this decision making process.  Here I will make several simplifying assumptions to make the problem analytically tractable.  These assumptions can be relaxed in practice albeit at a cost of having more complex equations.

In the simplest formulation, each person has an internal valuation $p$ of the ticket.  If the actual price $q$ is higher, this person does not buy the ticket. If the price is lower, this person buys the ticket at a rate (probability per unit time) proportional to the difference between the valuation and the price. The coefficient of proportionality is $1/\tau$, where $\tau$ is the characteristic time it takes to buy the ticket if the difference between the price and valuation is unity.  This $\tau$ can be called the customer response time.   I will make two simplifying assumptions: 1) $\tau$ does not change with time. This is not true in practice as people tend to make snappier buying decisions closer to the event; and 2) every person has the same $\tau$. Both assumptions can be relaxed by considering the response time that varies with time and is different for every individual.

The final ingredient in describing the population of people is the distribution of ticket valuations.  If the only information we have about this population is its size $L$ (total number of people potentially interested in the event) and the mean ticket valuation $p_0$, the number of people with valuations in a narrow interval between $p$ and $p+dp$ is $L/p_0 e^{-p/p_0} \, dp$, i.e. the distribution of valuations is exponential with a variance and mean equal to $p_0$.

Let me first consider the case of a fixed ticket price $q_0$. Also suppose that tickets go on sale some time $T$ before the event. The probability that a person with internal valuation $p > q_0$ buys the ticket during this time $T$ is

$\displaystyle{\mathcal{P}(p,T) = 1 - e^{-(p - q_0)T/\tau}}$.

Therefore, the expected number of tickets bought is

$\displaystyle N = \frac{L}{p_0} \int_{q_0}^\infty dp \, e^{-p/p_0} \, \mathcal{P}(p,T) = L\, e^{-q_0/p_0} \, \frac{p_0 T}{p_0 T + \tau}. \quad\quad\quad\quad (1)$

The gross receipts $N q_0$ (because everyone pays the same price $q_0$) are maximized when $q_0 = p_0$, i.e. when the price matches the mean valuation among the interested people. This is the best fixed price strategy when the expected number of tickets sold at price $q_0 = p_0$ is smaller than the capacity of the event. When the event capacity is smaller than the expected number of ticket that will sell at price $q_0 = p_0$, the price can be set higher to such a value that the expected number of sold tickets is equal to the event capacity.

The main conclusion is that when the ticket price is fixed and the event capacity is greater than $\displaystyle{\frac{L}{e}\frac{p_0 T}{p_0 T + \tau}}$, maximum gross receipts are achieved when the event is partially sold out.

The population of people potentially interested in the event is characterized by three parameters: 1) size $L$, 2) mean ticket valuation $p_0$, and 3) response time $\tau$. All of these parameters can be estimated by fitting the number of tickets sold as a function of time to the formula in Equation (1).

Having explained why partially sold out events could be the outcome of a rational pricing strategy we could call it a day and quit. However, pushing a little further turns out to yield something unexpected.

Let’s relax the assumption of a fixed ticket price. A declining ticket price which starts out quite high makes sense because people with a high ticket valuation will have likely bought a ticket early on and therefore more money can be made by expanding the pool of people who will buy a ticket at a lower price as the time of the event nears.

Let’s consider a linearly declining ticket price which starts high at $q_1$ when tickets first go on sale and ends up at $q_2 < q_1$ at the of the event. I will not include the formulas here as they are pretty messy. However, as shown in the figures below there is something interesting, namely a discontinuous transition.

To be clear, the results below are for the case when the event capacity is large enough so that the best strategy is not to sell out. It turns out that when the ticket sale duration $T$ is shorter than the characteristic customer response time $\tau$, fixed ticket price is the best strategy. However as $T$ becomes larger than $\tau$, the linearly declining price is better. To achieve the best result, the initial price has to be quite a bit larger and the final price quite a bit lower than the average ticket valuation. Therefore a careful measurement of $\tau$ is required to pick the best pricing strategy (fixed price vs. linearly declining price) because mid-flight switching between strategies is not feasible.

In real life, each person’s ticket valuation is not a constant and can depend on a lot of factors such as how soon the event is, promotions and the number of tickets already sold. A full model which takes all these factors into account would have to be constructed if precise quantitative predictions are to be made.

Gross receipts for two different pricing strategies as a function of the time $T$ before the event tickets go in sale. When $T$ becomes larger than the customer response time $\tau$, it is better to employ the declining price strategy. The mean ticket valuation for this graph is $p_0 = 2.$

When the ticket sale duration $T$ is larger than the customer response time $\tau,$ linearly declining price strategy is better than the fixed price strategy. The starting (high; red) and ending (low; green) prices are shown as a function of the sale duration.