Just to add some context for anyone looking to answer this, the authors cite a handful of major examples including the two World Wars and the Six-Day War. In WW1, starting around 1915, Britain and France ramped up their expenditures: Germany did starting around 1917, but only because it became clear that losing the war meant the government would be replaced. In WW2, they claim that Germany didn't really switch to a full wartime economy until the war started turning sour for them. And in the Six-Day War, they argue that Israel basically fought harder than its opponents, spending much more money per soldier. They also cite the two Gulf Wars as example.

The argument isn't only that autocracies sue for peace more quickly: it's that they don't fight as hard. This is supposedly because autocrats are more concerned with retaining their hold on power, which might or might not have anything to do with winning or losing the war, and the money they would otherwise spend on winning the war is better spent on paying people off (either literally or figuratively) so that they'll remain loyal. By contrast, in a democracy, leaders dedicate more effort to war because their constituents really do not like to lose, nor do they like their own soldiers getting killed, meaning that a lost or bloody war is likely to result in the rulers being removed from office come election day.

An excerpt follows:

With [our assumptions] in mind we can begin to unravel the seeming surprise of a larger military, backed by a larger gross domestic product—$5.3 billion derived from 30 million people in 1967 Egypt, compared to $4 billion generated by only 2.6 million Israelis—losing to a puny state. Imagine that the Israeli government spent as much as 10 percent of its revenue on private rewards, probably a high estimate. Imagine that the Egyptian government spent 30 percent of its revenue on private rewards; that is, more than the Israelis as befits the comparison of large- and small-coalition regimes that we have seen in the earlier chapters. Then how valuable did winning have to be for Israel’s coalition and for Egypt’s coalition to justify trying so hard that it meant spending extra money on the war effort?

Anticipating the high risk of war, the usually fractious Israelis formed a unity government in May 1967, reflecting the national commitment to win the coming war. We know the government allocated $381 million to the military in 1967. That means, given our assumptions, that $38 million of that pot of money might have been available for private rewards to the government’s winning coalition. Of course, even more would have been available across the whole economy (both in Egypt and in Israel) but we focus just on money committed to the military in 1967, thereby understating our case. Being a unity government it is likely that the Israeli winning coalition was very large, but we will err on the side of conservatism and assume that the government needed just 25 percent of the population to sustain it. That puts the winning coalition’s size at roughly 650,000 people. With these numbers in mind, we see that the potential value of private rewards taken from the military budget for government supporters in Israel would have been less than $60 a head ($381 million in military expenditures x 10 percent for private rewards/650,000 coalition members = $58.62 per coalition member).

Each member of Israel’s coalition could have had a choice: take the private reward or agree to put that money toward the war effort. Putting it toward the war effort would certainly have increased the odds of victory, an attractive public good to offset the small private gain that would be sacrificed by each individual in the coalition. Surely each of the relevant 650,000 Israelis would have put a greater value on military victory than a paltry $58.62!

Compare this calculation to that for Egyptians in Nasser’s winning coalition. We did a pilot study a few years ago in which we surveyed country experts about the size of several governments’ winning coalitions from 1955 to 2008. The experts we interviewed about Egypt placed its winning coalition as being as small as 8 members and as many as 65 in 1967. Wherever one comes down in that range it is obvious that the coalition was very small. We suspect the experts may have underestimated its size so we will err, again, on the side of conservatism and assume it was as many as 1,000 key military officers and essential senior civil servants. Even with our conservative estimate, each coalition member stood to get $150,000 in private rewards ($500 million in military expenditures x 30 percent for private rewards/ 1,000 coalition members = $150,000 per coalition member) if the funds out of the military budget that were available for that use were turned over to them instead of being applied to making a concerted increased effort to win the war. Whereas Israeli coalition members were only asked to sacrifice about $60 to help their country win the war, Egypt’s coalition members would have had to personally give up $150,000 in income to help their country win. It should be obvious that Nasser would likely have lost the loyal support of lots of his key backers if he took their $150,000 a head and spent it on the war instead of on them. He actually would have increased his chance of being overthrown in a military coup by making an all-out effort to win the war at the expense of his cronies. His backers would have had to place a value on winning the war that was worth their personally giving up $150,000. Victory is nice, but it probably isn’t that nice for many people. Levi Eshkol faced no such problem. His supporters were much more likely to place a value on victory that was greater than $58.62.