# Statistics for Dynamic Pricing of Theatre | Towards Data Science Summary: Dynamic pricing is the practice of adjusting a price to meet its demand, or market value. Given the difficulty of selling 100% of a performance’s tickets and that any unsold tickets immediately expire once a performance starts, dynamic pricing shows promise of lowering ticket prices while increasing revenue for shows. Included are several statistical simulations of pricing scenarios, along with implementable take-aways.

TERMS:

• Capacity: The maximum output that a company can produce. In theater, it is measured by percentage of seats filled.
• Stochasticity: Owing to random probability while independent of it, related to an underlying pattern or relationship. In this paper, the term is relevant for understanding the effect of scarcity on demand.
• Mean Reservation Price: The average price that a group is willing to pay.
• Mean Booking Time: The average amount of time before a performance when a group purchases their tickets.
• PDF (Probability Density Function): The probability that x will occur. The area of this function is always equal to 1 since the sum of all events will equal 100%.
• CDF (Cumulative Distribution Function): The probability that a variable taken at random will be less than or equal to x. Displayed graphically, the slope of a CDF will always move closer to 100% on the right, and closer to 0% on the left.
• Lognormal Distribution: A distribution used to measure continuous random quantities when the distribution is believed to be skewed with extreme values in one direction, such as people’s income.

SECTION I. Factors Affecting Theatre Ticket Purchases
What influences the purchasing of a ticket to theatre?

1. Price
The cost of a ticket is a serious consideration any potential attendee will consider.
In any microeconomic market, it is assumed that each potential customer has a reservation price of which they will pay below but not exceed. The distribution of reservation price is related to disposable income¹, thus behaving in a lognormal fashion.

To demonstrate, assume that the mean reservation price (mrp) of an audience is \$65 and the standard deviation is \$25.

If pricing economics were normally distributed, customer reserve price would be symmetrically distributed as shown in the below PDF (left). Since customers will complete a purchase if the offered price is below their reserve price, the percentage of paying customers will be the reverse CDF, (below right).

Notably, there are equal number low price customers as there are high. Thus, as the CDF moves from high prices to low, the % of paying customers rises in a symmetrical pattern above and below the median — which is also the inflection point.