An economic model of the mystery novel, and are sports suspense-optimal?

Those questions are considered by Jeffrey Ely, Alexander Frankel, and Emir Kamenica in their new JPE paper “Suspense and Surprise.”  Here is one to the point excerpt:

In the context of a mystery novel, these dynamics imply the following familiar plot structure.  At each point in the book, the readers thinks that the weight of evidence suggests that the protagonist accused of murder is either guilty or innocent.  But in any given chapter, there is a chance of a plot twist that reverses the reader’s beliefs.  As the book continues along, plot twists become less likely but more dramatic.

In the context of sports, our results imply that most existing rules cannot be suspense-optimal.  In soccer, for example, the probability that the leading team will win depends not only on the period of the game but also on whether it is a tight game or a blowout…

Optimal dynamics could be induced by the following set of rules.  We declare the winner to be the last team to score.  Moreover, scoring becomes more difficult as the game progresses (e.g., the goal shrinks over time).  The former ensures that uncertainty declines over time while the latter generates a decreasing arrival rate of plot twists.  (In this context, plot twists are lead changes.)

There are ungated versions of the paper here.  Note that at the very end of the paper…well, I’ll just let you read it for yourselves.

Comments

"We declare the winner to be the last team to score."

This is why the win-by-two rule is useful... tennis, volleyball, badminton (actually not sure about badminton, now that I think about it...). The winner must win the last point played.

Wait what? Doesn't the winner in such games win the last point played regardless of the win-by-two rule?

Hmmmm. I misspoke.

The final point must be a second consecutive point, not a 2 pt win. It does create suspense.

Right so. The reasoning of the two-point rule is that when something alternates (like serve, side of court, etc), a match won't be decided unless both competitors had the advantage the same number of times (or one more time but winning by two).

I think basketball would be vastly improved if after the 3rd quarter, we just added 20 points to the higher score, and said, first team to that score wins.

Or, for that matter, make it score based instead of time based. It's halftime when one team gets to 30, and the game is over when one team gets to 60.

It gets rid of all the fouling and time outs at the end of close games, and it means that it doesn't serve any purpose for the winning team to drain the clock. And, it means that a team that falls far behind has more of a chance to catch up - like in baseball.

I'd fix basketball by awarding possession to the fouled team rather than the fouling team. The idea that a team is rewarded for breaking the rules (and getting caught!) is crazy to me. Going even further (and here I will lose anyone who was previously with me), I wonder whether "breaking the law is the way to get ahead" is the best signal that we should be sending to the NBA-watching community...

I like your idea but I don't agree that fouling is breaking the rules. Cheating is breaking the rules. Intentional fouling is using the rules.

I think "foul" is pretty much synonymous with "breaking the rules." And I hate the idea that committing fouls should ever be profitable. Nevertheless I do see your distinction, and you could even strengthen it by noting the difference between ordinary and flagrant fouls...

In the rules of basketball, it states things that are not legal, like physical contact. Those are the rules. Breaking the rules results in turnovers like travelling and also in fouls.

So Ricardo is correct.

Lead changes aren't really like plot twists. The dumb conclusions are the logical result of a dumb premise.

Didn't mean that as a reply.

I might watch basketball under such rules.

First team to 100 wins... at least for the NBA.

As an added bonus, it would also mean that games would always end with a made basket - typically against an especially intense defense.

My own pet solution to the "first three quarters" problem in basketball (and I agree it is a problem), is to have the first team that wins three quarters win the game. You would start counting which team won each quarter. If a team wins each of the first three quarters, then win the game and the game would end early. If each team won two of the first quarters each, the game would be extended to a fifth quarter which would decide the winner. Obviously many if not a majority of games would wind up of normal length, with the loser winning one quarter of the first four, but they would be more interesting.

However, the idea of the first team to get to whatever threshold of points winning the game is a good one too.

Won't this move the fouling problem to the ends of each quarter then?

Good idea. The problem with basketball, as I see it, is that it has a very high rate of scoring until near the end, when the team that is ahead can drastically slow the rate down. So an 8-10 point lead, say, in the middle of the game can be overcome with strong play in a reasonable amount of time, but becomes a huge lead late in the game when the pace can be slowed to force the trailing team to foul.

Football, which has a vastly slower rate of scoring, switches this by incorporating devices to speed up scoring late - two-minute warnings, various clock stoppages, etc.

I'd like to see the trailing team retain possession after scoring, but some of the other suggestions here are good also.

"We declare the winner to be the last team to score. Moreover, scoring becomes more difficult as the game progresses (e.g., the goal shrinks over time)."

I think that would reduce scoring and drama greatly because the last team to score would stop trying to score again (no point in widening a lead that doesn't count) and instead go into a defensive shell that becomes easier and easier over time.

Although it would also mean there'd also be no penalty for pulling your goalkeeper when "behind."

PGI has correctly solved for the equilibium. A half-field game with leading team packing the box in defense.
Headline here is: JPE has found a way for soccer to have even fewer goals!

Yeah, this would really stink. Even now, when a team is ahead and starts to play defensively, the game can get (even more) boring. But under these rules there is absolutely no benefit to scoring again.

How about this: each time a goal is scored, it makes both nets get smaller. The last goal scored wins it. Then at least both teams would have an incentive to keep scoring. You could also start out with larger nets than exist now, increasing the overall scoring of the game.

Not that I'm really to worried about fixing soccer.

'We declare the winner to be the last team to score.'

Somebody hasn't heard of old-fashioned sudden death overtime, apparently.

Somebody hasn't actually thought about the post, apparently.

The point is attempting to eliminate boring blowouts (which sudden death overtime does not help).

What about having a random process call time, instead of a fixed time limit. That way even early leads are thrilling because there's a chance that the game could suddenly end. Even in a blowout the losing team would be on the edge of their seats early game, instead of just dejected. You could also vary the intensity or the process based on how close the score is, how long the match has been dragged out, or any number of parameters to tweak excitement and fairness.

This isn't altogether different than the model that Quidditch operates under. The snitch could be caught at anytime, so any lead is at any point is exciting. The problem Quidditch succumbed to was that broomstick technology got to good. Seekers started deciding every game before the quaffle points had any material impact on the final score.

Isn't that how soccer works... the clock runs out, and then they keep playing for some random interval....and then the ref blows the whistle.

Sure, but I'd propose the entire game fall inside the random interval.

A game that might be five minutes long is really hard to sell tickets to.

But the really big problem with any attempt to make a lopsided game more exciting is that it inevitably reduces the reward for skilled play. This may make a single game more interesting, but it makes the sport much less interesting. Most sports fans follow the big picture rather than a single game. It is about the whole process of the team being assembled and playing out a season. Signings, trades, injuries, and great or poor individual player seasons all add to the drama. The more essentially random game results are, the less this story matters. Even though an individual game might be exciting.

In other words, sports are optimised for actual fans, while papers like this are probably written by casual occasional game attendees.

How is harder to score over time any different from it being harder to mount a comeback with little time remaining?

This guy also misses the point that both the mystery reader and the sports fan want to see justice served in the outcome. If the score at the end of the half is 5-1 with the team with 1 winning because they scored last, you have a stupid game that nobody is going to watch or play. It would be like changing the rules in a mystery so that the guilty party is not who committed the murder but the last one who farted.

Perhaps scoring should always shrink the goal of your opponent. So instead of the goal shrinking with time, it should be be score. That way there'd still be an incentive to pile on even if you were already winning. But the right shrinkage curve (?) to use as well as how in the world to do that practically (hold on while the crew swops in a new goal frame!) still big problems to solve.

Also, one of the joys of watching live sports is to witness those very unlikely freak comebacks that form the sport history legends. I can remember when in ice hockey world championship semifinals in 2003 Finland-Sweden semi-final Finland lead by usually definitive 5-1, but Sweden came back and won 5-6 in front of the Finnish audience in Helsinki, making 3 goals in the last 2 minutes. That is something that is discussed every team Finland hockey game.

Giving any extra help to the team behind will always seem like cheating. That's why fans don't want to increase the randomness of games by that mechanism.

Some sports have natural feedback mechanisms that create suspense. In a running race, the guy coming second might be deliberately leaving something in the tank. Then again, the guy coming first might have enough in reserve to fend off that challenge. You can't know until the final sprint.

More subtly, in cricket there is a trade-off between scoring rate and risk. In limited overs games, this can cause the team batting second to try and win in the last few overs. Sometimes the strategy misfires and the game goes down to the wire.

I think beer pong may actually be the most suspense-optimal "sport" I have encountered. A friend of mine calls it a "game of convergence," since as you get down to fewer and fewer cups, your target zone shrinks and it gets progressively harder to sink one in. This gives the trailing team plenty of opportunities to catch up, and most games end up being pretty close even if there's a clear skill differential between the teams.

Not to be overly self-referential, but you could apply Hotelling here.

Lead changes aren't really like plot twists. For one thing, a game usually has a context within a larger season. A single game is more like a chapter than a book.

Sports has more suspense than any novel because you aren't on your gaurd as much. The game may be boring and therefore when it is exciting it is all the more exciting.

All games would feel the same under these dullards' rules.

Or you could have an 82 game NBA schedule that's so long that nobody could possibly try hard for the entire game so both teams just coast until the last six minutes.

Or a 162 game MLB season in which each game is so inconsequential that nobody cares about baseball until September.

Since an NBA half-time score is meaningless it would be better if their games were sudden death... from the very beginning. First team to score wins. That way they could play maybe 50 games in one evening, a 4100 game season.

The problem with mystery novels on paper is that you can tell how much of the story is left by the number of pages remaining, so you know that if all the evidence points to Colonel Mustard in the Conservatory with a candlestick but there are a hundred pages left, you shouldn't take that too seriously.

This is also a relative weakness of television series vs. mystery movies. If you're watching a episodes season of The Killing, you're pretty sure that the reveal in Ep3 is a red herring. The Affair had a pretty good formula to deal with this. The first half of the season, the mystery sub-plot was on a slow boil, focusing instead on interpersonal drama. Then by the end of the season the mystery took front and center.

Granted baseball has a pacing problem (but not really for us multi-taskers), but a game can still swing pretty drastically in the late innings, primarily due to wide variance in pitching performance. Any ace pitcher becomes vulnerable after three times through the line-up. The blown save situation. As the game progresses managers make irreversible substitutions that either maximize their chance of winning or are their undoing as their resources deplete.

MLB needs to enforce a pitch clock though.

And computer strike zones

+1 to pitch clock and computer strike zones.

I also agree that baseball is a good example of games being exciting in the late innings. Each team is always technically able to win until the game is over. Large lead changes are rare, but they do happen.

The example of the least exciting late game situation is in (American) football, when the team that has the lead can just kneel and run out the clock. However, admittedly the same clock system does lead to many exciting finishes.

Almost no sport in the entire world would be improved by making it pointless for a team to score twice in a row. In soccer or football or basketball, it would completely suck if the deal were "score, then convert to pure defense/run-out-the-clock."

But if you did that, why would either team bother with the game during the early parts of the game?

You'd have to also make there be uncertainty about when the game would end, otherwise might as well just rest up until the final quarter.

Every sport has a signal component, i.e. something that makes sure the better team wins more often than not, and a noise component, so element of randomness that prevents the better team from winning all the time. Every part of the game should be equally loaded on the signal component, that way the competitors are equally incentives to compete at every point. However the noise component should be loaded at the end of games, so the viewer has low certainty even up to the end. Hence creating suspense.

For example imagine in basketball if in the first half, shots counted two points. But in the second half any given shot would randomly count for eight points 25% of the time and zero the rest. Teams would still play equally hard in both halves because every shot has an expected value of two. But the point totals would swing much more in the back half of the game, so even a team with a big lead could lose.

Here boxing is exceptional, one reason for for the Rocky movies etc.

If we introduced tennis-style scoring to more sports, that would eliminate a lot of blowouts. Blowouts are boring.

The team that wins the most quarters wins the game. If it's 2-2, you play an overtime. The only downside would be that if one team won the first 3 quarters, there'd be no point in playing the 4th. If you want to keep the 4th, you could do something like require the team that's behind to double the score of the leader in that quarter.

Agreed on tennis-style scoring. Bring that to baseball. Not much of a fan of the keep the 4th strategy though. Nothing wrong with a game ending early, except for advertisers.

Other than that, I'd say there isn't much recognition of hockey rules. Richard Epstein recommended hockey-style penalties for soccer. Great idea.

The purpose of a sporting event is to determine the better competitor on that particular occasion, not to create suspense for spectators, which may or may not be incidental to the event. In fact, fans of teams or individuals want to see the object of their adoration crush the opposition. Nebraska football fans are unhappy if the Cornhuskers win by only two touchdowns. The Big Red coach was fired even though his record in the Big Ten was better than that of the Minnesota Gophers, whose coach was named Big Ten Coach of the Year.

The problem with this analysis is that it assumes that unpredictability is a synonym for suspense. Having games which have highly variable outcomes may lead to people being more uncertain, but may not lead to more suspense.

Take, for example, the recent NFL game between the Packers and the Seahawks. The Packers were up by 12 with only minutes left to play, a lead that should be insurmountable, but which wasn't. The Seahawks went on to win the game, which I can only assume that all non-Packer fans found to be one of the better games in recent memory. The history of the game established that a team in such a position would and should lose the game, failing to do so was a plot twist which in any novel would be unbelievable (in that it breaks the suspension of disbelief), but in sports was unbelievable (in the good meaning of the term).

If people really cared only about uncertainty, they'd stop watching football games after the coin toss, after all that's the most uncertain aspect of the game - nobody has any knowledge of how it will turn out. Instead, people care most when the predicable rules of sports intersect with the unpredictable nature of life.

You can get an idea of how this might play out by looking at sprints in track cycling.

I think people also want the game to be fair in the sense that the better team has the better chance of winning. When an obviously better team loses because of a "last goal wins" rule than the fans will get aggrieved and if it happens to often then they will stop watching. Just look as how upset fans get by a bad call from a referee/umpire when it effects the game.

In the Super Rugby competition, teams score game points during the game and from that they are awarded competition points - If a team wins the game they get 4 competition points, 2 points for a draw, 1 competition point for losing but being within 7 game points of the winning score, 0 competition points for losing and being more than 7 game points away from the winning score. And both teams can get another competition point for scoring 4 or more tries (equivalent to a touchdown) in a game. (Tries being an outcome that viewers like to see.) And teams get through to a knock-out stage by having the most competition points.

This system means the losing team keeps trying because they can get competition points by staying within touch of their opposition and by scoring tries. It doesn't mean blow-outs don't happen but it does reduce them.

How about Jeopardy style - dollar [ie. point] values are doubled in the second half.

A good first half (while everyone is still getting warmed up) can give you a valuable lead, but it's typically "still anyone's game" heading into Double Jeopardy. Heck, even the name seems fitting - the higher dollar values in the second half indicate the higher jeopardy (risk, uncertainty) that the competitors face. Keeps you on the edge of your seat.

You could even throw in a Final Jeopardy-esque element. Though as some have pointed out, the more heavily you tinker with these point systems to keep a game undecided until the end the more likely it is luck will contribute to deciding the winner.

Comments for this post are closed