# Preemptive bribes in legislature

Last week, Jeff discussed Deficit Chicken and the consequences of playing out the game. The August 2nd deadline is almost upon us, and up until now pretty much every product of debt-ceiling negotiations in Congress has failed to yield anything that has a shot at passing in both houses. How are the members of these parties “standing strong” together, anyhow?

It’s clear that some incentive prevents the moderates of each party from switching over — that’s why there’s a Gang of Six and not a Gang of Sixteen or Twenty (yet). Whether it’s pride, pork, or pressure, as Ted DiBiase would say, “Everybody has a price.”

This price, for an individual legislator, would be equivalent to the utility of standing his ground. Let’s suppose these utilities look like this for nine representative legislators on a committee. (Notice that I represent utility below the line as a negative utility from the status quo — it is a positive utility for the opposite outcome.)

Suppose the other party has a budget of 20 (say, in billions of dollars) to spend on bribing members of this party to join their coalition. The status quo party (call it X) has the first move and can pre-emptively give the committee members some additional utility for staying on this side, and afterward, the other party (Y) has the opportunity to make counteroffers.

Now, say that Y needs to ultimately win over 5 of these 9 committee members. As it stands, who would Y bribe? Certainly, Mr. I is already on Y’s side, so Y needs to bribe four more people — probably the least expensive of them, E, F, G, and H. Perhaps party X should devote its energy toward bolstering these weak members, possibly even try to win the dissenters back.

But, this isn’t always the case. Tim Groseclose and James Snyder in their celebrated 1996 paper offered a delightful best response strategy for party X in their vote-buying model.

To win over 5 members in the current state, Y just needs to spend 4 + $\epsilon$ on E, 3 + $\epsilon$ on F and G, and 2 + $\epsilon$ on H to win. Party X can make Y’s life harder by preemptively giving E, F, G, and H more money, say 1, 2, 2, and 3 respectively, to make their payoffs 5 each — this exhausts Y’s war chest. But now, there is a less expensive party member (D) for Y to bribe!

So, X must give all of the members (except I) enough to have a payoff of at least 5, so that there are no “soft spots” to target and Y cannot win over any of them. It turns out that the best strategy for X is always this kind of “leveling schedule,” and X often ends up bribing more members than Y needs to win. We can easily test this here: suppose the optimal strategy is not a leveling schedule. If we add x more to anyone’s payoff, it is unnecessary. If we remove x from anyone’s payoff in the leveled coalition (say, we bribe H with 2 instead of 3). Then, Y will certainly target H, and have 20-4 = 16 left to spend on winning 3 more votes. X would need to add 1/3 to each of C, D, E, F, and G, to prevent Y from winning any three of four, which is another leveling schedule (and one that happens to be more expensive than the 1 saved from H.)

It could be, then, that “standing strong” for our status quo party really does mean emulating Aesop’s fable of the bundle of sticks (which, incidentally, is fascio in Latin).