THE BOOK--Playing The Percentages In Baseball

What about the other players’ chances of getting the question correct?

Wouldn’t your odds go up a bit by simply risking all but $1, assuming both other players risked it all?  How much would that change it?

Oops.  Guess I skipped the last Para.

In Jeopardy nowadays winners come back in perpetuity so the expected winnings for the future if you win today is not $45K but 45+15+5+… That makes winning today even more important than a big payoff today.

Chris G. - the people behind almost never risk it all for the simple reason that they hope the player(s) above them get it wrong and bet enough that they end up behind them.

If you bet $25 000, then the break-even point for “given that I missed the question, what chance does #2 have at getting the question right?” is 33.3%.  I’d wager that the chances are worse than that.

On that note, I’ve often wondered whether there is an unwritten rule against betting the exact amount to tie instead of winning if you are leading.  You’ve shown that you’re dominant over the 2nd-place finisher (yeah, yeah, sample size, you got lucky hitting favorable categories), so why not drag him along to the next day?

Somewhat off-topic:

Say you’re the $40,000 guy in the example above, and the Final Jeopardy category is something besides baseball, of average difficulty. The other 2 contestants have $15,000.  Why not bet an even $10,000?  You’re guaranteed to come back another day, and there’s a possibility another contestant (who is presumably an inferior contestant, albeit in a SSS) comes back to join you on the next show.  I suppose there’s prestige in being the day’s champion, but I’d prefer to come back facing at least one opponent I’ve more or less defeated.

There’s other factors here of course—the returning opponent would be familiar with the surroundings, the buzzer, the pressure, etc., while the brand new contestants are unfamiliar. It’s just something I’m surprised to see happen so rarely.

Toffer:

Let’s say the average winner makes 30,000$ for each game, and you have a 1 in 3 shot of winning.  And if you win, you get to come back the next game.  So, one shot in 3, you pocket 30K and come back again.  Two times in 3, you are left empty-handed.  Your overall expectation therefor is 45,000$.

That’s 30 + 10 + 3.33 + ..., or 30/(1-1/3) = 45

Ties: I don’t follow Jeopardy.  Are you saying that in the case of a tie, both contestants come back the next day?  And they both keep 100% of their winnings (they don’t split it?).

Also, do the losers get 0, or the amount they are left with?

Yes—if two (or three) contestants tie for the game’s high score, they both return the next day while keeping 100% of winnings.

Losers get 0, but receive around $1000 consolation to help defray costs of travel/hotel (the show does not pay for either).

It’s not always about risk aversion. It’s sometimes a matter of non-linear utility. Every dollar isn’t equally as valuable as the next. The first $10k in my pocket is truly more useful than a marginal $10k when I already have $35k.

#8

Losers also get prizes.  Not great, but 2nd place prize is better than the third place prize.

Let’s say the average winner makes 30,000$ for each game, and you have a 1 in 3 shot of winning.  And if you win, you get to come back the next game.  So, one shot in 3, you pocket 30K and come back again.  Two times in 3, you are left empty-handed.  Your overall expectation therefor is 45,000$.  That’s what playing in a 30,000$ game in Jeopardy is worth, if the winner gets to keep coming back.

Am I missing something?  Doesn’t that one chance in three mean that your expectation of future winnings on the 2nd day is $15,000 (1/3 of $30,000 + (1/9 * $30000)+ etc.), not $45,000?

I liked Richard’s take in the Yahoo! comments:

In terms of betting strategy, here’s my take: If you’re willing to bet that your baseball knowledge is better than either of your competitors, bet JUST enough money that, even if you lose, you’ll still have $1 more than what the 2nd place person had BEFORE the final question. Chances are, if you’re gonna get the question wrong, so are they… but you don’t wanna bet the farm on an impossible question, just so that the software engineer who knows nothing about baseball, and therefore bet $0 on the final question, wins by default.

Ouch, my bad.  It was 30K for current earnings and 15K for future earnings.  That’s why I had 45K.

Ditto what Peter said, I was wondering if anyone was going to catch that.  That doesn’t change the BE point, though, I don’t think.  I always thought that playing for the tie was the way to go, and that you should discuss that with the other contestants before the game starts.  I suppose, though, that you are prohibited by the show to make any private deals with the other players.

Some people on the show who are really good, have around an 80-90% chance of getting Final Jeopardy correct, in my estimation (just a guess).  Yet they almost always take the risk-averse strategy when they are way ahead to guarantee the victory.  That always pisses me off, especially when they have a lot of money going into the final round and they are way ahead.  For example, if they have 30,000 and the next closest has 13,000.  They will invariably bet 3,999 even if they are very confident in the category, say 80%.  That is chickenshit!

Using Tango’s math, that is an expectation of 47,399.40 if they bet 3,999 and 60,000 if they bet the whole thing, and that doesn’t even take into consideration the fact that they may still win if they bet 29,999 and the other 2 players go broke (which usually doesn’t happen but it could, maybe 5% of the time, adding another 750.00 to his expectancy).  I’ll take the extra 12,600 in expectancy any day of the week, but I am a gambler!

Then again, there is one other consideration.  The player that wins one day has a much greater chance of winning the next, I assume.  So rather than 1/3 it might be 40 or 50%.  Also, if you are really good, like a Jennings, surely you want to make sure that you keep playing since you have like an 80% chance of winning the next day.  Also, the better you are, the more money you are likely to win when you win again.  Of course, if you think that your win was a fluke, all the more reason to go for as much as you can in that one game!

Anyone know how often the winner wins again?  That should be fairly easy to look up, I would think.

BTW, my cousin Allan Lichtman, who just ran for the Senate Primary in Maryland and lost, is an undefeated champion on Tic Tac Do!  I went for a group interview for the same show about 20 years ago in NYC and even though I scored well on the initial test, I was never called back for the second round. Not sure why.  Maybe they sensed that I was arrogant and self-righteous.  When I watch Jeopardy on TV I usually do well on the first two rounds and I am awful on Final Jeopardy. I always figured that if I got on that show, I would have to double the next highest contestant before the last round in order to win.  Basically, I would be going for broke on all the daily doubles, although they tend to be hard too.  I have read that one of the most important things is to be able to negotiate the buzzer, and of course answering questions from the comfort of your own home, with no one else to out-buzz you and with no pressure, is a lot easier than having to do it for real…

Whoops, looks like I was reading too quickly and looked at the wrong numbers. But I guess that happens even to the best of us.

Well, I got 8 out of 10, and I am not very good at baseball history at all.  I missed the Boston Red Stockings one and the expansion team one, although I might have gotten that one given enough time.  In 30 seconds under pressure, I would have probably gotten only 6 or 7.

Well, I’ve been there, and though the category wasn’t baseball, I risked too much and lost (being in second place).  From that perspective, and hindsight is 20:20, I’ll say this:

1.  Ignore the category and play the safe route (i.e., bet no more than $9999).  Although if you actually think you have the second place person’s number, I think the full $10,000 bet is better, because why not have an opponent in the next batch you think you can beat rather than two wild cards?

2.  The returning champion has a huge advantage in terms of motor skills on the buzzer.  You get just a couple of minutes in rehearsal to play with the buzzer (and it’s not game conditions, it doesn’t help much), compared to someone who has had it in his hands for a full round or more (and obviously did it well since he or she won).  When they did the Super Champions round where Ken Jennings was guaranteed a spot in the final, he was nowhere near as facile with the buzzer as he was in his lengthy prior appearance, while his opponents had played a large number of rounds to make the final.  He was coming in cold and got whupped.

And the only question I missed in the baseball examples was the second one, using Jeopardy time.  But trust me, it’s different under the lights.

Bread, so, are you allowed to make an agreement with your opponents before you start that you will bet “$10,000” (or whatever) so that there is a possible tie rather than playing for the $1 win.  If you can make that agreement beforehand, that hugely benefits all players.  Even without the agreement, I would think that there would more plays for ties, especially if you think your opponent was not that good, or just because it is the “gentlemanly” thing to do, even if there was no prior agreement.  I have never seen anyone intentionally play for the tie - maybe once.

So what was the final category and question that you missed?  How much did you and your opponents have going in?  Or you don’t want to give away your real identity?

MGL - I think your reasoning perfectly explains why you are an economist/mathematician/gamlber. While your actions would be perfectly Economically LOGICAL they would be not be HUMAN. While going for broke may net you greater expected rewards financially, people think and decide on many more factors than simple financial reward. As was stated earlier, the marginal benefit of the first $10K is much greater than the 7th. On top of that there are also other factors to consider: pride (would you really be happy if you pulled a Cliff Clavin?), the ability to tell a good story to friends, family, coworkers, etc; the enjoyment of playing multiple days, etc.

Using a simple net benefit calculation in a situation such as this is quite frankly, simplistic.

As one of the other posters mentioned, I would have bet $25,000.  I’d be willing to risk more than $10,000, but I’d have to guard against a question that everyone get’s wrong.

I got only 8 out of the 10 baseball final jeopardy questions right.  Looks like I may be overconfident and that’s within the television pressure.

"As was stated earlier, the marginal benefit of the first $10K is much greater than the 7th.”

That entirely depends on how much money you have or what you plan on doing with the money. For example, even if you had virtually no money and your goal/plan was to purchase a BWM 6-series car that costs around 70K, then the marginal benefit of the last 10K on your way to the 70K is greater than the first K, or at least equal.  So let’s be careful with the blanket statement that the first 10K has greater marginal benefit than the 60th or 70th K.

Another thing. It depends on whether this is your only (or one of a very few) brush in life with this kind of a choice.  For someone who gambles for a living, let’s say a skilled professional poker player, every hand presents this kind of decision.  If the he consistently follows the risk-aversive strategy, he will earn a lot less money in the long run and perhaps be a loser rather than a winner.  The only thing that matters to the serious professional gambler in MOST cases, is the mathematical expectation of his decision.

So let’s also be careful with equating risk-averse strategies with being “human.”

A classic example of this phenomenon is if you ask a typical non-gambling person whether they would rather take a guaranteed $100,000 or flip a coin and if it lands heads he gets $250,000 and if it lands tails he gets nothing, most ordinary people who are not wealthy (and even plenty who are wealthy) will choose the former.  No self-respecting gambler who has some money will make that same choice!

MGL, any form of collusion will violate the agreement you sign when you get on the show and void all payments. 

I have no particular need to hide my identity here (you have my email).  This is my game:

http://www.j-archive.com/showgame.php?game_id=1965

There’s a lovely graph at the bottom of the link that will show you what I meant about the buzzer; I knew nearly every answer after the point where I never rang in first again.  So did either or both of my opponents and they beat me to the buzzer and never missed again (except one or two no one knew).  21 right, zero wrong, missed Final Jeopardy!, got a great free trip to Aruba (worth more than the champion’s winnings).  My 22 minutes of fame, seven more than my Warhol allotment.

I agree that we can’t assume that marginal utility of money decreases. That’s a common assumption but it has basically no empirical basis (at least w.r.t. individuals). There are lots of cases we can come up with where it doesn’t hold, like MGL’s BMW example.

Or say you will die in one day unless you get a $1,000,000 operation. Clearly the marginal utility of money /increases/ as you get closer to that goal (and the utility of the last $1 is very large indeed).

"So let’s also be careful with equating risk-averse strategies with being “human.”

A classic example of this phenomenon is if you ask a typical non-gambling person whether they would rather take a guaranteed $100,000 or flip a coin and if it lands heads he gets $250,000 and if it lands tails he gets nothing, most ordinary people who are not wealthy (and even plenty who are wealthy) will choose the former.  No self-respecting gambler who has some money will make that same choice!”

Like I said that is why you think like a gambler/economist. My point about being human is that 95% of people will choose differently. Humans simply act much differently than the cold, rational beings that economists expect them to do. In fact they are usually acting quite rationally when you consider the context. As you suggest, gamblers face these decisions all of the time but most people do not, and that’s going to affect their decision-making.

Using the same example as you, if I gave you $10 million dollars or the 50/50 chance at more than that what would you break even point be? Would you truly answer $20M + 1 cent?

Can you really find me a person with virtually NO money who’s only goal/plan is for a 70K BMW? Would they REALLY turn down guaranteed money for the chance at the full 70K. It’s nice hypothetical but it just doesn’t really happen in real life.

I’ve only seen a few episodes of Deal or No Deal but contestants are constantly presented with the choice of guaranteed bank money (which is always less than their net average expected return) or going for it. While they always push their luck for a little bit I never saw one go for broke. I’m sure it happens (particularly if the big money is taken away early and the stakes are small) but I bet that it is very rare.

MGL #21:

How about if you have to bet a dollar, and have a 1 in 10,000,000 chance of winning $15,000,000.  That’s an expected gain of $0.50 on every dollar played.  You’d have to be really mathematically ignorant not to bet everything you had, right?  Or, how about a 1 in 1 billion chance of winning $1.5 billion.  You’re a gambler, and you have lots of self-respect: would you spend every penny you have on such a bet?

If not, then at some point you decide it no longer makes sense to do it.  I think it might be because at some point the huge payoff becomes more than a person could possibly need, while the buy-in becomes more than a person could possibly spare, but that’s just my take on it…

MGL #21:  you have to come up with a pretty contrived scenario to change the shape of a utility curve convex. There’s a ton of work on this in psychology and economics with some Nobel prizes to validate it. It’s safe to say that any contestant on Jeopardy will value $40,000 more than a 50% chance of $80,000 or a 75% chance of $60,000. How much more? Only they know. What you call risk-aversion is the rational strategy for maximizing utility, not winnings.

I think that there are two separate questions for the risk aversion discussion: 

1) Is a given person’s utility function sufficiently non-linear in the region in question to justify taking the safer option.

2) Is that person irrationally averse to gambling and therefore unwilling to accept a wager which will be to his benefit on average even considering his utility function.

In the extreme examples (e.g. involving positive expectation bets with a tiny chance to win billions), non-linearity will affect almost everyone, including those experienced at assessing wagers.

In more mundane cases, irrational aversion to risk or values based aversion to the appearance of gambling is a bigger player.  In trying to understand attitudes towards risk, I’ve talked to upper middle class people who were horrified at the thought of making a positive expectation wager where their worst case loss was roughly the same size as the daily standard deviation of their stock portfolios.  The difference seemed to be that owning stock is “investing”, not “gambling” and seen less in terms of consciously taking risk.

Bread, I didn’t see your name in your email address!  So you had the interesting situation where you can only win if the player in first misses the last question.  So you have the choice of betting nothing, betting at most $8,600 (to make sure you win if player I misses and player II gets it right and bets her whole stack), or something in between.

If you think you have greater than a 50/50 chance at the right answer, you bet $8,600, if you think your chances are less, you bet zero.  If they are 50/50, it doesn’t matter what you bet.

The other interesting thing you did, which I rarely see on that show, was that you played for the tie with player III.  Given that she apparently was not a very good player, that was a brilliant play!

Too bad you missed the question.  I got it right, even though I usually suck at Final Jeopardy, but then again, I am from NY, and it was a little bit of a guess!

Greg #25:

If you look to maximize your expected bankroll then yes you have to bet everything you have on either of those bets.  But I think that most gamblers who think explicilty about issues like this look to maximize the expected growth of their bankroll, which gives rise to the Kelly Criterion for bet sizing.

This would dictate a bet of 3.3x10^-8th of your bankroll on the first bet. So if you had a million dollar bankroll you would bet about three cents on such a bet (and if the minimum bet were one dollar the expected growth of the bet drops below zero and you do not play).

The Kelly criterion suggests a bet of 3.3x10-10th of you bankroll on the second bet.

A Kelly criterion-based bet sizing on the Final Jeopardy question would give a different result than the Tango’s bet-sizing suggestion, which is based on maximizing the expected value of your final bankroll rather than the expected growth of it.

"The only thing that matters to the serious professional gambler in MOST cases, is the mathematical expectation of his decision.”

Serious professional gamblers care about maximizing *growth* more than expected *value*.  This is a subtle but important distinction that I’m sure MGL is aware of, but I think it warrants being explicitly mentioned.

Also I think that in MOST cases, serious professional gamblers do not encounter situations where they are betting with more than than net worth.  Not only are there different psychological considerations there, but even the mathematics changes.

I didn’t see Dave Allen’s post when I made mine, but I was basically talking about the Kelly criterion.