Randy Cox
United States Clemson South Carolina
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We played this game last night and it was moderately enjoyable. However, there is one thing I can't get my head aroundthe ability to skip over certain levels of the Pole Vault and High Jump.
I understand why athletes do that in the real Decathlon. It's a way to conserve precious energy for future events, particularly future events on that same day of competition.
But in terms of this game, why would you skip over a particular height, except to speed up the game? I could understand it if there were a penalty for excess attempts, but I don't see any in the rules.
My suggestion would be that if you take over a certain number of total attempts (say, seven) in the High Jump, you lose a reroll in the 400 meters for each further attempt. And in the Pole Vault, do likewise with rerolls for the upcoming 1500 Meter Run.
Thoughts?

Andrew Brannan
United States Rockville Maryland

Randy Cox wrote: We played this game last night and it was moderately enjoyable. However, there is one thing I can't get my head aroundthe ability to skip over certain levels of the Pole Vault and High Jump.
I understand why athletes do that in the real Decathlon. It's a way to conserve precious energy for future events, particularly future events on that same day of competition.
But in terms of this game, why would you skip over a particular height, except to speed up the game? I could understand it if there were a penalty for excess attempts, but I don't see any in the rules.
My suggestion would be that if you take over a certain number of total attempts (say, seven) in the High Jump, you lose a reroll in the 400 meters for each further attempt. And in the Pole Vault, do likewise with rerolls for the upcoming 1500 Meter Run.
Thoughts?
In game terms, it's a way to risk losing at a higher height, rather than have to hit two heights. For example, let's say that you have a 25% chance at height X and a 20% chance at the next height Y, figuring in all rerolls, etc. If you go for both, you only have a 5% chance of making Y points, but 25% of the time you'll at least score X points. Or you can skip X and go straight for the 20% chance of Y points, but 80% of the time you'll end up with no points. It's all about risk vs reward.



There's good reason to skip certain heights in high jump: simply, because if you attempt every height, you increase the risk of being knocked out of the competition at an earlier stage. As soon as the bar gets to 18, it starts becoming difficult to succeed. The odds of successfully jumping an 18, and a 20, and a 22, in succession, are quite low. Remember that a roll of 24, when attempting to clear a height of 18, only counts as a clearance of 18. So why not save that roll of 24 for a bigger height?
I used this strategy to win the high jump event in a recent decathlon competition described here: New World Records: Celebrating the 2008 Olympics with Knizia's Decathlon http://www.boardgamegeek.com/thread/335793
The other competitors all attempted every height, starting at 10, and going up in increments of 2. They all cleared 10, 12, 14, and 16, but then one was knocked at 18, two were knocked out at 20, and one at 22. I however, elected to skip 10, 12, 16 and 20, and successfully cleared 14, 18, and 22, and was unable to jump 24.
As you can see, in this case, the strategy worked. I attempted relatively easy 14 just to ensure I would get some points for the event. Should I have attempted 16 and 20? Had I rolled the 22 when attempting a height of 20, for example, I would have only scored 20, and then very likely I would have failed to have cleared 22 as well. On the other hand, there is a risk: had I rolled a 20 when attempting 22, I would have only scored the 18 from my previous jump, since I'd elected to skip the 20.
So the strategy is somewhat risky, but what it counts on is that the rewards outweigh the risks in the long run. By being able to skip heights just like in real life, and doing so for strategic reasons, I think this event is another instance of where the theme of this game fits the mechanics very well!

Steve Cates
United States Visalia California

to strain the example say your first roll is going to be (5)6's would you want that to be at 10 feet or 30 feet. If you want it to be 30 feet, pass all the heights.

Larry Welborn
United States Anderson South Carolina
Clemson Tigers #1

Randy Cox wrote: We played this game last night and it was moderately enjoyable. However, there is one thing I can't get my head aroundthe ability to skip over certain levels of the Pole Vault and High Jump.
I understand why athletes do that in the real Decathlon. It's a way to conserve precious energy for future events, particularly future events on that same day of competition.
But in terms of this game, why would you skip over a particular height, except to speed up the game? I could understand it if there were a penalty for excess attempts, but I don't see any in the rules.
My suggestion would be that if you take over a certain number of total attempts (say, seven) in the High Jump, you lose a reroll in the 400 meters for each further attempt. And in the Pole Vault, do likewise with rerolls for the upcoming 1500 Meter Run.
Thoughts?
I like the idea. Decathlon also penalizes a competitor for taking too attempts.

Randy Cox
United States Clemson South Carolina
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ironcates wrote: to strain the example say your first roll is going to be (5)6's would you want that to be at 10 feet or 30 feet. If you want it to be 30 feet, pass all the heights. But all these examples (including previous posts) treat the roll as something sacred. "You'd hate to lose that great roll..." doesn't make sense statistically. The odds of rolling a 22 when trying for a 14 is exactly the same as the odds of rolling 22 again when trying to clear 22. There is no difference.
You might roll 28 on every single clearance from 10 to 28.
Anyway, my problem is that there is no penalty for excess jumps, therefore nothing stops a player from trying them all. No real decathlete would take three tries at 10, 12, 14, 16, 18, 20, 22, and 24 if they routinely clear 16 or 18. They's save their energy.
But in this game, there's no energy to save and no penalty for wasting it. So you might as well try to clear every height. No skin off the athlete's nose if it takes three full tries for every single height. So what if someone jumped over 6 feet 24 times in a row? But in the real Decathlon, they's be crazy to do that and then try to run 400 meters.
Anyway, statistically speaking, a player would be crazy not to try every height. There is nothing to be gained in skipping a height, except time.

Randy Cox
United States Clemson South Carolina
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abrannan wrote: In game terms, it's a way to risk losing at a higher height, rather than have to hit two heights. For example, let's say that you have a 25% chance at height X and a 20% chance at the next height Y, figuring in all rerolls, etc. If you go for both, you only have a 5% chance of making Y points, but 25% of the time you'll at least score X points. Or you can skip X and go straight for the 20% chance of Y points, but 80% of the time you'll end up with no points. It's all about risk vs reward. Mathematically, if there is a 25% chance to clear height X and a 20% chance to clear height Y, there is no reduced chance to clear them both.
When you're rolling for the first height, you have a 25% chance to score those points. Either you make it or you don't. Your results are 100% or 0%, nothing in between.
Then if you made the first jump, on your second jump, you still have a 20% chance to clear it. Not 5%. As long as you make the easier jump, you still have the same chance you had before. In other words, whether you skip or clear, you'll have a 20% chance at height Y.
You're talking about the chance of clearing both before attempting the first. But since these things are done in sequence, it's a Bernoulli and we don't need to concern ourselves with the second jump. If we can't make the first jump, we surely couldn't have made the second jump on that roll. If we make it, the slate is clear on our next jump.



Randy Cox wrote: Anyway, statistically speaking, a player would be crazy not to try every height. There is nothing to be gained in skipping a height, except time. I'm afraid I'm with Andrew Brannan on this one, and I believe his statistical analysis correctly proves that there is something to be gained by skipping a height.
To try to explain it another way: let's say that you have a 25% chance at clearing 20, and a 20% chance of clearing 22 (I haven't calculated the exact odds). Method 1 (Brannan + Ender): only attempting 22 i.e 20% chance of success. Method 2 (Randy Cox): first attempt 20, then 22 i.e. you still have a 20% chance of success when you attempt the 22, but you will only get the opportunity to attempt the 22 if you first clear the 20, which is only a 25% chance. So you are reducing the odds of overall success in making 22. The odds of clearing 22 on its own are the same (20%), but you're reducing them by first requiring that you clear 20 (25%), which will only happen 1 in 4 times. So the odds of clearing both heights is 1/4 of 20% = 5%! Clearly, Method 1 is superior.
If you're not convinced, try playtesting, and I think you'll find that playtesting will replicate the experience I documented above more often than not. I'll leave it to the statistics and math gurus to prove this further.
Edit: Some more anecdotal evidence  in a decathlon just completed by three of my children, they immediately attempted heights of 28 and 30 in the pole vault without any previous jumps, and so successfully set a new world record in this event, by using Method 1. My daughter in fact rolled a 36  had that been the height she had attempted immediately, she would have cleared it; and the odds of successfully doing this are much higher than Method 2, which would first require successful rolls of 28, 30, 32, and 34 etc before undertaking a difficult 36.

Lexingtonian
United States Unspecified Massachusetts

To boil it down, Randy, you seem not to be accounting for the fact that a low, unsuccessful roll incurs less of a penalty if it follows a successful roll for a relatively high jump.
Of course, if you skip a height, you lose the opportunity to score at that height, but which risk to take is up to the player to figure out strategically.

Randy Cox
United States Clemson South Carolina
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Let's use the hypothetical figures above where no one attempts anything up to the 20/22 level.
Player B+E has an expected value (return on their investment, so to speak) of 0.2 * 22 = 4.4.
Player R has an expected value of 0.25*20 + 0.05*22 = 5+1.1 = 6.1.
Over the same number of games, Player R will tally more points than Player B+E.
Of course, boiling things down to the math takes away from most of the appeal of the game. It should be a shootfromthehip game. Though I do wish there were a penalty for excess jumps to incite more skipping.

Dane Peacock
United States Stansbury Park Utah
That tickles

I was a pole vaulter in high school, and while I do not know what the rules say in Reiner Knizia's Decathlon game, I do know that there is a very good reason to skip heights in an actual pole vault (or high jump) event. It all has to do with ties. Ties are broken as follows:
1. The tiebreaker goes to the competitor with the fewest number of tries on the last cleared height. 2. If the tie still remains, the competitor with the fewest total number of unsuccessful trials throughout the competition, up to and including the height last cleared, shall be awarded the higher place.
Both of these tie breakers come into play nearly every meet because multiple vaulters go out at the same heights and points are awarded through six places.
To be eligible, the vaulter must have at least one successful attempt. So, the vaulter does not want to wait for a height that he struggles to make, but if he is confident at clearing the early heights, then it is a wise decision to skip them because of the risk of adding unsuccessful attempts that will count against him in case of a tie.



Randy Cox wrote: Let's use the hypothetical figures above where no one attempts anything up to the 20/22 level.
Player B+E has an expected value (return on their investment, so to speak) of 0.2 * 22 = 4.4.
Player R has an expected value of 0.25*20 + 0.05*22 = 5+1.1 = 6.1.
Over the same number of games, Player R will tally more points than Player B+E. Ahh, I think I see where you're coming from now. 1. Player R has the greater risk of being eliminated more quickly than Player B+E. But: 2. Player B+E has the greater risk of not getting any points at all. So over multiple games, the expected outcome will be that Player R will tally more points than Player B+E, but Player B+E will get more world records than player R.
For example, over multiple games, the scores in this event for both players might look like this: Player R: 16, 18, 12, 14, 14, 20, 14 Player B+E: 0, 0, 22, 0, 0, 24, 0
In short: 1. For best chance of getting some points: attempt every height (low risk, low reward). Disadvantage: higher risk of early elimination. 2. For best chance of getting high points: skip lower heights and attempt only higher ones (high risk, high reward). Disadvantage: higher risk of scoring no points at all.
Should players attempt every height? Not necessarily. It becomes a balance of juggling risk and reward. If you want a larger reward, skipping some heights may be worthwhile  but it comes with a larger risk.
Possibly a combined strategy is the best, by jumping a low height in order to get some points on the board, and then skipping some heights from time to time. Then the scores in this event over multiple games might look like this: 14, 14, 22, 14, 0, 24, 14
But ultimately, this is a decision that is left up to the players: do they want a low risk low reward strategy and attempt every height, or a high risk high reward strategy by skipping some heights? For example: "my opponent is attempting 18, shall I skip 18 and just attempt 20, hoping that I'll succeed with 20 and that he'll either get eliminated at 18 or 20?" These are the kinds of decisions that make the game interesting.
Just like in real high jump, conserving energy by skipping some heights in order to avoid elimination can be a viable strategy under the right circumstances.

Randy Cox
United States Clemson South Carolina
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EndersGame wrote: For example, over multiple games, the scores in this event for both players might look like this: Player R: 16, 18, 12, 14, 14, 20, 14 Player B+E: 0, 0, 22, 0, 0, 24, 0 I just don't see it. Player R has just as much chance of getting a 22 or 24 as player B+E does. Any time.
The key is that you don't have a stockpile of "good rolls." You don't "waste a good one" just because you clear the 12 bar with a roll of 28. You still have just as much chance of rolling 28 on the very next roll (though you need only 14).
But if you do roll poorly (like, 11 or something), it doesn't matter if you are rolling for a 14 or if you skipped all the way to 30. You fail either way.

Lexingtonian
United States Unspecified Massachusetts

But the more dice you roll, the more chances there are to fail, and one failure puts a stop to it.

Randy Cox
United States Clemson South Carolina
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No. Taking more turns doesn't increase your chance of failure. If you are going to roll only 9, you're not going to advance regardless of the level. Whether you're trying at 10 or skipping to 30, you fail equally.
If you succede, then great. You have pocketed some points and are in exactly the same position you were in before your prior turn. You've lost nothing by passing that hurdle.

Matt Crawford
United States San Francisco California

I had the same question as Randy when I read the rules. This is a weird discussion. Randy's right, there is no advantage to skipping heights. But sometimes probabilities are hard to explain and harder to get. So if Randy's arguments haven't convinced you yet, I don't know how either.
I think that some people conceive of dice rolling as "I am going to roll what I am going to roll. I can make judgments about strategy after seeing the rolls," like Ender did with his children's game.
It's like in Yahtzee  if you roll 66662, and you say, "I won't keep any. I'll roll them all again." Then you proceed to roll 55555, was that a good decision? Obviously not (to a probabilist) because your chances of improving on your 6666 roll were awful!
If only Reiner would come and weigh in, then people would beeleeve!

Bill Herbst
United States Sayville New York

Randy Cox wrote: No. Taking more turns doesn't increase your chance of failure. If you are going to roll only 9, you're not going to advance regardless of the level. Whether you're trying at 10 or skipping to 30, you fail equally.
Having read many other things about statistics by you on BGG, Randy, my instinct on this subject would generally be to trust you, but it seems to me as if you're in error here. Taking more turns does not increase your likeliehood of failure for a particular roll  you're absolutely correct in noting that a roll of 9 would fail regardless of the height of the jump. The point is that the more attempts one makes the greater the odds of rolling such a number. Would you agree that there is a higher probability of making a single roll at a height of 26 than there is of rolling each combination starting from a height of 12 and ultimately reaching the roll at 26? The second option involves achieving successful rolls 8 times in a row. You may well be correct in noting that if you're simply looking to maximize your acheived height, you should attempt every height. Playing the game for honors though in which only the top three scores count, I would probably skip some early heights in a larger group because the odds of everyone crapping out at a height of 12 or 14 are very low. If I were interested in achieving a medal in a group of 8 players (a likely situation I will face tomorrow night), I would probably start jumping after the low heights. At least in that way I could avoid completely crapping out on a height that will almost certainly not feature in determining who will be in the top three spots at any rate.



An experiment
The math behind this whole question is fascinating me, and it was enough of a challenge for me to try an experiment. I quickly produced 200 results with five dice using a java dice roller found here: http://www.shadowflux.com/dice.html
I then examined those 200 results, to see what would be the outcome for a player beginning at 14, and (using the same dice results) the outcome for a player beginning at 16, and so on.
The outcome
Results looked like this, for the different starting heights: 14: 22,16,22,14,18,18,20,22,20,20,18,20,18,22,24,20,20,14,22,16,16,20,20,18 16: 22,18,22,16,18,20,20,16,22,20,18,18,18,22,20,22,22,20,20,0,20,16,20,18,18,18,20,20,20 18: 0,22,20,22,18,18,20,18,20,22,22,20,20,18,20,20,18,22,22,22,22,0, etc 20: 0,22,0,20,24,0,0,20,0,20,20,20,0,22,22,22,0,0,20,20,0,20,0,22,0,22,22, etc 22: 0,22,0,0,26,0,0,0,0,22,22,0,0,0,22,24,0,22,0,0,0,22,0,0,0,0,22,0,22,24,24,0,0, etc
Analysing the data
starting height of 14 (sample size: 24) highest of 14: 2 = 8.3% highest of 16: 3 = 12.5% highest of 18: 5 = 20.8% highest of 20: 8 = 33.3% highest of 22: 5 = 20.8% highest of 24: 1 = 4.2%
starting height of 16 (sample size: 29) highest of 0: 1 = 3.4% highest of 16: 3 = 10.3% highest of 18: 8 = 27.6% highest of 20: 11 = 37.9% highest of 22: 6 = 20.7%
starting height of 18 (sample size: 33) highest of 0: 4 = 12.1% highest of 18: 7 = 21.2% highest of 20: 12 = 36.4% highest of 22: 10 = 30.3%
starting height of 20 (sample size: 43) highest of 0: 17 = 39.5% highest of 20: 12 = 27.9% highest of 22: 13 = 30.2% highest of 24: 1 = 2.3%
starting height of 22 (sample size: 51) highest of 0: 32 = 62.7% highest of 22: 13 = 25.5% highest of 24: 5 = 9.8% highest of 26: 1 = 2.0%
Now to compare these results, to see whether there is any effect on the chances of scoring high or low.
Percentage getting 18 or higher: starting at 14: 79.2% starting at 16: 86.3% starting at 18: 87.9%
Percentage getting 20 or higher starting at 14: 58.3% starting at 16: 58.6% starting at 18: 66.7% starting at 20: 60.5%
Percentage getting 22 or higher starting at 14: 25.0% starting at 16: 20.7% starting at 18: 30.3% starting at 20: 32.5% starting at 22: 37.3%
Conclusions
What does this mean? Clearly the sample size is not large enough to draw definite conclusions, especially because the results when starting at 14 have the biggest margin of error. But some tentative conclusions:
Conclusion 1: starting at 14 or 16 only slightly decreases the odds of getting 20 or higher. This is simply because the percentage of jumpers that are eliminated at 16 (3.4%) and at 18 (12.1%) is relatively small, so only about 15% of those who start at 14 are eliminated before they can attempt 20.
Conclusion 2: starting at 14 or 16 strongly decreases the odds of getting 22 or higher. 3.4% are eliminated at 16, 12.1% at 18, and 39.5% at 22, i.e. about 55% are eliminated before they can even attempt 22. Of those who attempt 22, only 37.3% succeed. Ergo: the lower the starting height, the lower the eventual reward.
Conclusion 3: starting at 14 or 16 and jumping every height strongly decreases the odds of getting 24 or higher. 11.8% of those who started at 22 ended up jumping 24 or higher. Of those who started at 14, 16, 18, and 20 and jumped every height at increments of 2, there was only two other successful jumps of 24, and none of 26. Clearly, for the greatest chance of jumping 24 or higher, one should begin at 22 or up and skip the earlier heights. Ergo: the higher the starting height, the higher the possible reward
Conclusion 4: starting at 14 or 16 decreases the odds of scoring a zero. The more heights one skips and the higher one begins, the greater the chance of scoring no points at all. Ergo: the higher the starting height, the higher the risk.
Extrapolating using an illustration
To extrapolate from these results, using an example: imagine two groups of 100 jumpers. Group 1 (Brennan & Ender) begin at 20. 60.5 of the 100 make it. Group 2 (Randy Cox) begin at 14, and jump every height. 3.4% fail at 16, leaving 96.6. 12.1% fail at 18, leaving 79.1. 60.5% of these 79.1 make it, i.e. only 47.9 jumpers are left who successfully clear 20. So of Group 1 (Brennan/Ender), about 60 of the 100 jumpers make 20, but of Group 2 (Cox), only about 48 of the 100 jumpers make it! Admittedly, the 40 jumpers in Group 1 (Brennan/Ender) who failed have 0 points, whereas the 52 jumpers in Group 2 (Cox) who failed 20 all have 14, 16 or 18 points. But doesn't that confirm my point about the correlation between risk and reward?
Overall conclusion
Granted, these conclusions are based on a limited anecdotal sample of results. But they do appear to support my earlier hypothesis that there can be good reason to skip heights. As stated previously, by skipping heights, there is the potential of greater reward (eg scoring 24 or higher), but it comes with greater risks.
I respectfully suggest that this experiment goes some way to providing data in support of the hypothesis posited by Andrew Brennan and myself.
The strategic implications: a) Low risk, low reward: Starting at 14 and attempting every height, will yield at least 14 points, but is less likely to score 24 or higher. b) High risk, high reward: Starting at 22 or 24 is much more likely to yield nothing, but is also the most likely way of achieving a score of 24 or higher.



EndersGame wrote: Randy Cox wrote: Anyway, statistically speaking, a player would be crazy not to try every height. There is nothing to be gained in skipping a height, except time. I'm afraid I'm with Andrew Brannan on this one, and I believe his statistical analysis correctly proves that there is something to be gained by skipping a height. To try to explain it another way: let's say that you have a 25% chance at clearing 20, and a 20% chance of clearing 22 (I haven't calculated the exact odds). Method 1 (Brannan + Ender): only attempting 22 i.e 20% chance of success. Method 2 (Randy Cox): first attempt 20, then 22 i.e. you still have a 20% chance of success when you attempt the 22, but you will only get the opportunity to attempt the 22 if you first clear the 20, which is only a 25% chance. So you are reducing the odds of overall success in making 22. The odds of clearing 22 on its own are the same (20%), but you're reducing them by first requiring that you clear 20 (25%), which will only happen 1 in 4 times. So the odds of clearing both heights is 1/4 of 20% = 5%! Clearly, Method 1 is superior.
For 100 jumps:
method 1: 80times score 18, 20times score 22.
method 2: 75times score 18, 20times score 20, 5times score 22.

Randy Cox
United States Clemson South Carolina
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OK, I see where we differ on this. I think in absolute terms of every single attempt at a jump. On any attempt, you have no better or worse chance of clearing the height whehter you skipped some before or not. The dice have no memory.
What y'all are saying (I get it now) is that you want all the glory (big points, no whammies), regardless of the potential to royally screw up. I guess there is merit in that, if you're looking for world records.
I ran the numbers, too, for 200 progressions from 10 through 30 (yes, 2000 "jumps" requiring 6000 rollsthank goodness for Excel).
The first thing I discovered is that, expectedly, you can safely always start at 14. Never did I fail on the 10 or 12 jump. I also never had a success at 30. So, here are the breakdowns between 14 and 28.
Average Score (number across top is start height) 14 16 18 20 22 24 26 28 19.5 19.0 17.9 14.2 8.1 5.2 1.3 0.4
This is enough evidence for me, one who wishes to maximize score, to try every height, at least starting at 14 (or 16, if using my variant rule about attempts beyond the 7th costing you rerolls on the next running event). I mean, I could start at 22, but I'd be cutting my average score more than in half.
Success Rate (% of scoring at least something) 14 16 18 20 22 24 26 28 100 97.0 89.5 68.0 36.0 21.5 5.0 1.5
Glory Hog Numbers (% chance for each final score) 14 16 18 20 22 24 26 28 0 0.0 3.0 10.5 32.0 64.0 78.5 95.0 98.5 14 3.0 16 10.0 10.0 18 22.5 22.5 23.0 20 36.5 36.5 38.0 43.5 22 17.0 17.0 17.5 18.5 28.0 24 5.5 5.5 5.5 5.5 7.5 20.5 26 0.5 0.5 0.5 0.5 0.5 1.0 5.0 28 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.5
So, if you want to go for the glory, I guess it makes sense to start at about 22. You'll fail almost twothirds of the time, but when you succede, it will be glorious.
Be The Best That You Can Be (highest score registered under each scenario) 14 16 18 20 22 24 26 28 26 26 26 26 26 26 26 28
But in then, while you may have better chances at the glory numbers by holding off, the max (world record) seems to top out at 26 unless you're foolhardy enough to start your quest at the 28 level.
Conclusion For me, I just can't see passing on any attempts except at levels 10 and 12. Everything else just seems to me that the return diminishes too fast and the big payoff is so infrequent that you lose the war in deference to the single battle. I guess it all comes down to whether you're trying to win the individual game, or excel at the individual event and set a world record.



Excellent work Randy, you just got yourself a tip!
This is exactly the kind of statistical overview I was looking for in support of my hypothesis, and certainly an improvement on my own effort in my previous post. I'm glad we've been able to come to a common understanding! Thanks for providing this helpful data analysis, I think it satisfactorily addresses all our questions, and conclusively demonstrates that decathletes need to decide whether they want to maximize their overall score and their average score for this event by jumping conservatively, or be a glory hog in a risky quest for an individual event world record. Nicely done!

Werner Bär
Germany Karlsruhe Baden

Randy, i think the difference is that other people talk about skipping levels, not starting at a high level.
In your calculation, you assumed that people starting at a high entry point and gain 0 points when failing. But they would start at 14 for the guaranteed minimum points, and skip some of the intermediate levels.
Whether skipping is worthwile or not depends on the probabilities of a success at each level. Let's compare two different probability tables, and see what results each strategy has:

Expample 1: odds to succeed at 14 16 18 20 100% 70% 10% 0%
Player 1 does all tries, and has a result of: 30%: failed at 16, for 14 points (chance to continue: 70%) 63%: failed at 18, for 16 points (chance to continue: 7%) 7%: failed at 20, for 18 points. Average score: 15.54
Player 2 doesn't try at 16, and has a result of: 90%: failed at 18, for 14 points (chance to continue: 10%) 10%: failed at 20, for 18 points Average score: 14.4

Expample 2: odds to succeed at 14 16 18 20 100% 25% 20% 0%
Player 1 does all tries, and has a result of: 75%: failed at 16, for 14 points (chance to continue: 25%) 20%: failed at 18, for 16 points (chance to continue: 5%) 5%: failed at 20, for 18 points. Average score: 14.6
Player 2 doesn't try at 16, and has a result of: 80%: failed at 18, for 14 points (chance to continue: 20%) 20%: failed at 20, for 18 points Average score: 14.8

If the odds for each high are declining fast, the strategy to try each high is clearly superior. If there are several tries with low but similar success rates, skipping tries can be better.
When i use your numbers for a success chance at each level:
Randy Cox wrote: Success Rate (% of scoring at least something) 14 16 18 20 22 24 26 28 100 97.0 89.5 68.0 36.0 21.5 5.0 1.5 My spreadsheet tells me that skipping height (16), 22, and 26 has nearly no effect at the final result. If the odds for heigth 22 was 35.9% instead of 36%, skipping it would already be the better choice. Skipping other heights gives you a worse average result.

Cameron McKenzie
United States Atlanta Georgia

Werbaer wrote: Skipping other heights gives you a worse average result.
The folly here is that going for the highest average is not always the best way to win. If your down by a lot, taking risks is the way to go.
Suppose your opponent is doing slightly better than you. If you both go for "average" results, he will continue to be doing slightly better than you and you'll lose. If he goes for an "average" result and you take a risk to go "all or nothing" for this round. If you win it, you'll pull ahead and win the game. If you lose it, you'll fall behind a LOT and lose.
The thing is that for many people, losing by a lot is no worse than losing by a little. You have to go for the win even if doing so may make you lose worse. But it really depends on the person. For some people, their final score is what is important, regardless of how the others do. For other people, it's win or lose. The final score doesn't matter, just whether it was the highest one or not.

Randy Cox
United States Clemson South Carolina
1024x768 works just fine  Don't Wide the Site!
Missing old BGG

If we use a hybrid option, where a player locks in their 14 by attempting at that height, and THEN skips until a certain point with no other skips, then the average points changes considerably:
16 18 20 22 24 26 28 19.5 19.4 18.7 17.1 16.2 14.2 14.2
And that makes for a different argument. Given this, once you lock in at 14, there isn't all that much difference until you skip to 22 or 24, so I'd probably still go forward with everything from 20 onward.
But back to my original point. There is still no Decathlonrelated reason NOT to try every level (or 14 and then every level from 20 onward). So I can make a couple of jumps at 14 and then two or three at 20 and then three at 22 and three more at 24. If I'm lucky at that point, I could spend three more at 26 and fail anyway. That example would be 13 or 14 attempts, and that's a lot for a decathlete.
To encourage people like me (maximize points! maximize points!) to skip every now and then, I still think that a rule that says "for every jump or pole vault attempt beyond the 7th try, reduce your number of rerolls in the next running event by 1, not to drop below zero". I think that will add extra strategy to the game.

James Fung
United States San Diego California

MasterDinadan wrote: The folly here is that going for the highest average is not always the best way to win. If your down by a lot, taking risks is the way to go. I'm just about done with my Decathlon programmed opponent where I optimized for average score at every decision point. For the High Jump, the averageoptimized decision is to jump at every height. Randy's numbers are slightly off from the exact probabilities since he used a Monte Carlo estimation, but they're close and the analysis doesn't change. The probability of making heights starts to drop off quickly around 20 (the averageoptimal player starts with expected value 19.3; after achieving 18, his expected value is only 19.9), so you want the highest safety net possible.
Pole vault is almost the opposite. These are the probabilities of making each height:
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 86.4 83.8 77.8 75.0 69.3 66.0 60.7 57.2 52.6 49.1 42.8 33.6 22.9 12.9 5.9 2.0 0.5 0.1 0.0 0.0
The averageoptimized opponent attempts the following jumps:
Height Dice Value 12 4 17.3 28 8 20.6 32 8 29.5 36 8 32.5 38 8 36.1 40 8 38.0 42 8 40.0 44 8 42.0 46 8 44.0 48 8 46.0
I.e. he starts the event with an expected score of 17.3. If he can make height 12 (almost 1 in 6 chance of not making it), his expected score jumps to 20.6, mainly because he skips every height between 12 and 28. He can make 28 about half the time; he probably would not get to 28 if he attempted one or more intermediate heights. After 36, he starts attempting every height, but the probabilities are so low that he probably will not get much further.
The reason for this is that even low pole vault heights are not sure things, so attempting them is a risk. That said, the probabilities drop of more slowly than in the high jump, so it's better to skip several and not tempt fate.
As someone posted above, optimizing average score may not be optimal from Decathlon perspective, especially if what you care about rank. This is because a 20 is not worth twice what a 10 is worth. However, I optimized using the expected score as my metric because it's quick and good enough for a basic system.
By the by, my programmed opponent is done, but I can't post it yet because the table needed for 3 of the events (discus, javelin, long jump) are huge and incomprehensible. The other 7 are done and written up, but I was holding back until they're all done because releasing it. I have Python code and Excel spreadsheets if anyone is suffering from insomnia.


