程阳：美国强力球与超级百万单价变化后分析

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Power Ball and Mega Millions since the Price Change

Current sales in Power Ball are higher than would have been expected for similar jackpots in the $1 Power Ball game by about 11%, when the jackpot is below $250 million. Once the jackpot exceeds $250 million, sales are dramatically higher. Current sales of Mega Millions are the same as would have been expected for similar jackpots before the price change in Power Ball.

Background

Since January 2012, Power Ball (PB) and Mega Millions (MM) have been differentiated on price, and similar in many other ways. The odds of winning the top prize are very similar in the two games. The likelihood of winning a prize worth less than $200 is 3.1% in PB, and 2.5% in MM. The million-dollar second prize in PB is distinctive.

Since January 2012, MM has developed one record-setting jackpot of $656 million, and PB has twice developed notable jackpots of over $300 million.

WA Lottery Research and Development has established math models of national sales describing how players spent money in response to jackpots in the “post cross-sell, pre-price-change” base period, i.e. from March 2010 through December 2011. These models provide a baseline against which to describe changes in spending since the price increase. That is, they project what sales would have been for a jackpot of particular size on a particular draw day, during the base period. The discussion below is based on comparing our nationwide actual experience to these base models.

With only nine months of experience, it may be early to make generalizations about how things have changed. The record MM jackpot may have changed player expectations of both games. Consequently, the remarks below are based on the period after that jackpot.

Results

In the months since the record jackpot, MM sales were first a little higher and later a little lower than expected under the base model. The chart 1 shows this trend. During this period of time there has not been a MM jackpot over $118 million. The summed value of departures from the model, April through mid-October, amounted to only about 0.1% of the expected total value. In other words, MM sales on the whole have not been hurt, but have been very much as in the base period.

As shown in Chart 2, during the same April through October period, PB sales have been steadily higher than the base value. Aside from the two draws with jackpots over $250 million, PB sales were about 111% of the expected value. The PB jackpot of August produced “excess” sales of $107 million in the two draws where the jackpot exceeded $250 million. Total sales amounted to about 169% of the expected value.

The point of breaking out the two “exciting” draws separately is to emphasize that this is a very non-linear business. The price of a wager may matter more when the jackpot is in a “normal” range than when it is perceived as “high”. Building the jackpot from its base level is aided by an 11% increase in cash input, compared to the base game. The real benefit appears once the threshold for excitement is reached. Spending for the $320 million jackpot on 8/15/2012 was 184% of what would have been expected in the base period.

At a national level, the short story is that PB has been helped and MM has not been hurt or helped by the price increase in PB. As suggested by the national research study carried out in 2010, “brand loyalty” was such that players did not move their money very much in response to which game appeared to offer the “better deal” that week.

At the level of an individual state, the effect may be different. In Washington (a “MM legacy” state), for instance, PB sales agree almost exactly with the base model except when there is an exciting jackpot. That is, there is a benefit to Washington only when the PB jackpot develops beyond a critical value.