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Licensed Unlicensed Requires Authentication Published by De Gruyter September 6, 2013

Ticket Pricing and Scalping: A Game Theoretical Approach

Nelson Sá and Evsen Turkay

Abstract

Ticket scalping is frequently related to the economic puzzle of underpricing by promoters. It is also disputed whether event promoters benefit from scalper participation or not. Our article explores two questions: can promoters benefit from scalpers’ activities and what are the resulting consumer welfare effects? We address these questions by developing a three period game where the secondary market is supported by an auction mechanism, interacting with primary market decisions. We find that participation by scalpers can lead to underpricing in the primary market and that this may benefit small or credit-constrained promoters. This requires the scalper’s discount factor to be higher than the promoter’s discount factor. The necessary premium on the discount factor increases with the fraction of early buyers and decreases with market size. Finally, the effect of scalper participation on aggregate consumer welfare is shown to be positive for a large enough market size or discount rate for the scalper.

Appendix

Proof of Proposition 1

In the absence of the scalper, and . This implies that there will be units in the market after the first period. If the scalper participates in the market, he must buy this residual quantity. We want to show that the scalper has no incentive to participate in the market under these conditions.

If he participates, his profits can be computed using the objective function [5]. This yields

[22]
[22]

Given that , for all . The scalper will then not participate in the market with and .

Proof of Proposition 2

Take a quantity and assume that is the profit maximizing price for the promoter. The equilibrium definition implies that the scalper chooses such that his profits are non-negative. For a contradiction, suppose that the scalper’s expected profits are strictly positive. This implies

[23]
[23]

where denotes the unconditional expectation of the scalper. By definition and the promoter can increase his profits by choosing and keep the scalper still buying. Hence, a pair cannot maximize the profits of the promoter when the scalper participates in the market unless constraint [9] is binding.

Proof of Proposition 3

Eq. [12] may be rearranged as

[24]
[24]

From here,

[25]
[25]

Similarly,

[26]
[26]

Next,

[27]
[27]

Finally, the ambiguous sign of may be confirmed by simple numerical simulation. This holds over a wide range of values for N, and .

Concerning the number of units bought by the scalper, eqs [9] and [11] yield

[28]
[28]

Next,

[29]
[29]

with

[30]
[30]

and

[31]
[31]

As long as , . Since , it follows that . Direct inspection of eq. [28] gives .

Using eq. [11], the equilibrium number of early buyers is

[32]
[32]

This can be rearranged as

[33]
[33]

Direct inspection of this expression makes it clear that .

Proof for scalper participation condition

Suppose that eq. [15] is met for a set of values . Taking by reference [13], it is straightforward to show that

[34]
[34]

and

[35]
[35]

It follows that whenever condition [15] is met, so must be the non-negativity constraint [13].

Proof of Proposition 4

The right-hand side in eq. [15] is always positive. The left-hand side must also be positive for this condition to hold given an appropriate value for N. Next, notice that when , the left-hand side in eq. [15] equals . This is always negative for any . Furthermore, the left-hand side in eq. [15] decreases with a lower . It follows that when , it becomes increasingly negative. In conclusion, eq. [15] can only hold with .

Next, eq. [15] may be rearranged as

[36]
[36]

It is straightforward to show that the right-hand side in eq. [36] is decreasing in N. Also, because both terms are increasing in , so does the right-hand side.

Proof of Proposition 5

Using eqs [16] and [20], welfare can be shown to improve when

[37]
[37]

Expression [11] shows that . The sufficient condition is derived by assigning a value of to the first term and a value of 0 to the second term.

Acknowledgements

We are grateful to Geoffrey Jehle and Robert Rebelein for their helpful comments and suggestions. We also thank Jiayi Bao for her excellent research assistance. Any errors remain our own.

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  1. 1

    The distinction between a broker and a scalper is straightforward. The former is an officially licensed business, whereas the latter acts informally, frequently on an individual basis. Nonetheless, their conduct is fundamentally identical in that both seek a profit through secondary market exchanges. For the remainder of the article, we shall treat both agents in the same way.

  2. 2

    New York State laws illustrate this point well. Prior to 1984, resale for more than two dollars above face value was illegal in New York. This limit was gradually increased until the state introduced a 3-year experiment in 2007 completely deregulating ticket resale.

  3. 3

    In a study of 103 rock concerts held in the summer of 2004, Leslie and Sorensen (2009) show that most events offered tickets at only two price levels. Ideally, the promoter would like to employ price discrimination to its fullest possible extent, precluding the scalper from exploring any arbitrage opportunities. Transaction costs in operating secondary markets are one possible reason why this does not always happen. Negative reputation effects might also discourage the promoter from engaging in excessive discrimination.

  4. 4

    For an analysis of how a monopolist can explore individual demand uncertainty in ticket markets, see for instance Courty (2003b).

  5. 5

    Krishna (2009) shows that first-price and second-price sequential auctions yield the same results for sellers. Besides, they both entail the same level of analytical complexity. Weber (1983) also shows that simultaneous and sequential auctions with risk-neutral bidders holding independent private values yield the same expected payoff as long as there is no time discounting within the auction period (the third period in our model).

  6. 6

    Heterogeneous discount rates for buyers might enable their endogenous allocation to primary and secondary markets. However, it would not be possible to uniquely associate each group to one particular type of discount rate, much less with uniformly distributed sub-populations, thus compromising the analytical tractability of this model.

  7. 7

    It should nonetheless be noted that resale dynamics are likely to vary depending on whether a single event (our example) or a season ticket are involved. The ability to resell is frequently more valued in the second case since the ticket holder may not be able or willing to attend every individual game.

  8. 8

    This vital principle may be found in Gold v. DiCarlo, 235 F. Supp. 817 at 820 (S.D.N.Y. 1964). There, resale prices regulation is deemed constitutionally acceptable because it affects the price the public is forced to pay, thus making it a proper venue for the exercise of state police power.

Received: 2013-02-28
Accepted: 2013-08-20
Published Online: 2013-09-06

©2013 by Walter de Gruyter Berlin / Boston