# Party Tricks

## Twenty dollar bills and campaign finance reform.

One of my favorite classroom tricks is to auction off a $20 bill. I explain in advance that the $20 goes to the top bidder, but the top two bidders have to pay whtatever they bid. That way, if at least two students bid at least $10, I come out ahead.

In fact, it's far better than that. As soon as *two* students are rash enough to enter the bidding, I'm almost guaranteed an astronomical profit. Once Mickey bids a nickel and Minnie bids a dime, both are trapped: There's never a right time to drop out of the auction. If Mickey quits now, he loses his nickel, but by staying in he can quite reasonably bid a quarter for a $20 bill. Minnie, of course, raises her bid to half a dollar. After a few rounds, Mickey has bid $9 and Minnie goes to $10. Mickey can't quit without throwing away $9; instead he takes the far more sensible route of bidding $11 to get $20. Now my profit is in the bag.

But that's only the beginning. After a few more rounds, Minnie bids $18 and Mickey bids $19. Now Minnie can either bid $20 to get $20 or drop out and pay $18 for nothing; she bids $20. At this point, poor Mickey can either bid $21 to get $20 (accepting a $1 loss) or drop out and pay $19 for nothing; he opts for the lesser of two evils and bids $21. Similar reasoning leads Minnie to bid $22. The spiral stops escalating only when one or the other runs out of money or exhausts my willingness to extend credit. If my students were sufficiently wealthy (and sufficiently shortsighted to enter the auction in the first place), I could earn a lifetime's income in a 50 minute class period.

My auction game is a crude but instructive metaphor for political campaign spending. It might even be a reasonably accurate metaphor if we lived in a world where the biggest spender always wins. In any contested election, candidates would continue to spend until they had exhausted all their own and their supporters' resources. Foreseeing that outcome, no politician would ever contest an election. The first candidate to file would always win by default.

But in our world, outspending your opponent doesn't always make you a winner--it only increases your *probability* of winning. And beyond a certain point, it doesn't increase your probability by very much: The candidate who runs 10,001 commercials has little advantage over the candidate who runs 10,000. So, sooner or later, politicians reach a point where one more commercial just can't justify its cost. That puts a natural limit on what they're willing to spend. Mickey will always bid another dollar to get a $20 bill, but he won't always bid another dollar to get a slightly increased *chance* at a $20 bill.

All this is important because campaign finance reform is important, and you can't predict the effects of campaign finance reform unless you think about the incentives it creates for politicians. Let me offer a few potential scenarios.

First, imagine the most simple-minded of all possible reforms: The government subsidizes each of the two major candidates to the tune of, oh, say, $1 million apiece. You might expect that plan to backfire by encouraging the candidates to spend more. But think again: The subsidies don't affect the cost of the 10,001^{st} commercial, and they don't affect its benefit--so if a candidate has already decided not to run that commercial, he still won't want to run it. Therefore, total expenditures don't change--they just get covered by the taxpayers instead of the fat cats. (The only exception is if the candidate is literally out of money and never had the option of running the commercial without the subsidy.)

Here's a different scenario: Suppose that instead of giving $1 million apiece to the top two candidates, the government agrees to give a million to *any* candidate who meets certain criteria--say, a 10 percent showing in major opinion polls. Now the main effect is to encourage new candidates to enter. With more candidates must there be more total expenditure? Not clearly. Suppose, for example, that Al Gore and George W. Bush are in the middle of a hotly contested race for the presidency when Colin Powell suddenly throws his hat in the ring--and suppose that Powell has a 20 percent chance of winning. Then Gore and Bush are no longer battling each other for the presidency; they're battling each other for an 80 percent *chance* at the presidency. With the value of the prize diminished, they'll be willing to spend less in its pursuit. So what Powell spends is at least partly offset by what Gore and Bush choose to save.

That example points to an important general principle: Total expenditure is determined by the value of the prize, whether we're talking about presidential campaigns or state lotteries. If the Powerball jackpot is worth $50 million and less than $50 million worth of tickets have been sold, then the odds are favorable--and rational betters will rush in to buy tickets, guaranteeing that at least $50 million in tickets *will* be sold. Likewise, if the presidency is worth $50 million and there are many potential candidates with essentially identical chances of winning, they'll keep entering the race until they've collectively spent at least $50 million.

In a state lottery, your odds of winning depend only on how many tickets you hold compared with everyone else. That's what guarantees that people will keep buying tickets as long as the odds are in their favor. That's a good analogy to an election with many equally plausible candidates whose odds of winning might depend only on how much they spend compared with everyone else. But if some candidates tower over others, the analogy breaks down. If you know that the state lottery is likely to be rigged, you'll buy fewer tickets. And if politicians believe that an election is "rigged," in the sense that the favorite will probably win *regardless* of expenditures, they'll buy fewer campaign commercials.