Game Theory Games

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Squigs44
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Re: Game Theory Games

#21 Post by Squigs44 » Sat Aug 10, 2019 5:05 am

If my instructions for country selection were not entirely clear, here is an example:
Alice submits this list: FTGIEAR
Bob submits this list: TGEFIAR
Caroline submits this list: FIGAETR
David submits this list: FTEIARG
Edward submits this list: TGEFIAR
Fiona submits this list: GITFAER
Greg submits this list: TIEGAFR

In the first round, Fiona is the only person to have submitted Germany, and she is awarded Germany. There are three way ties for France and for Turkey, so those countries are randomly selected. Bob is randomly selected for Turkey, and David is randomly selected for France.

In the second round, 4 players remain. Alice, who chose Turkey 2nd, is bumped to the third round, since Turkey is already assigned to Bob. Edward, who chose Germany 2nd (which is already assigned to Fiona) is also bumped. Caroline and Greg both chose Italy as their second option, and it is still up for grabs. Italy is given to one of the two of them randomly. Greg wins the coin toss and gets Italy.

In the third round, there are just 3 players remaining. Alice and Caroline both picked Germany third, so they get bumped to the 4th round (since Germany was already snagged by Fiona). Edward, on the other hand, picked England third, and so he gets assigned England.

In the fourth round, Alice picked Italy, and so she is bumped to round 5. Caroline picked Austria though, so she is awarded Austria.

Since Alice is the only player remaining, she is assigned the last country - Russia!

This means that Alice only has a 3% chance of winning, while David has a 30% chance. Had Alice been selected to play France in round 1 though, she would have had the 30% chance, and David could have ended up with the 3% chance.
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Re: Game Theory Games

#22 Post by BobMcBob » Sat Aug 10, 2019 6:36 am

Hooray!
Squigs44 wrote:
Sat Aug 10, 2019 5:02 am
2 BobMcBob A Kiel 48.57
2nd place! I was originally planning to build A Bur (don't ask), but there seemed to be too much interest. So I switched to A Kiel, and it worked surprisingly well! Congratualtions to RoganJosh for winning and to gimix for choosing my location and forcing me into second :-D Well played.
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Re: Game Theory Games

#23 Post by Claesar » Sat Aug 10, 2019 1:53 pm

I PM'd Ancient Memories some days ago and predicted to them we had selected the exact same placement. Nailed it! Too bad I didn't predict the other 4.

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Re: Game Theory Games

#24 Post by Claesar » Sat Aug 10, 2019 2:05 pm

Do I understand correctly: In your simulations, I will be entered into 1,000,000/p games, where p is the number of players. In each game I'll have 6 random opponents. My preference will be pitted against each of those 6. Then each game is simulated and I may randomly win with some skewed odds.

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Re: Game Theory Games

#25 Post by Squigs44 » Sat Aug 10, 2019 3:20 pm

Claesar wrote:
Sat Aug 10, 2019 2:05 pm
Do I understand correctly: In your simulations, I will be entered into 1,000,000/p games, where p is the number of players. In each game I'll have 6 random opponents. My preference will be pitted against each of those 6. Then each game is simulated and I may randomly win with some skewed odds.
Yes. Except the number of games you play is approximate depending on how to fairly distribute games, and is closer to (1000000/p)*7, since there are 7 players in each game.
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Re: Game Theory Games

#26 Post by Squigs44 » Sat Aug 10, 2019 8:40 pm

Someone suggested a secondary contest of guessing the winner's solo%. I'll take guesses for anyone that wants to do that as well.
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Re: Game Theory Games

#27 Post by AncientMemories » Sat Aug 10, 2019 10:33 pm

Should've actually gone to con, i guess :)

Also, the winner of the game is the one who gets the most solo's? Maybe there should be two winners, whoever gets the most solos and whoever has the highest average expected solo rate? Might be more complicated than it's worth, but for an extreme example, an extraordinarily lucky player who always puts russia as number one might win the game as I understand it if they happen to hit the 3% sweetspot an improbable number of times. Or do I misunderstand the scoring? Obviously that's unlikely, and the intent for the 1e9 games is to reduce the chances of something like that, but...

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Re: Game Theory Games

#28 Post by Squigs44 » Sun Aug 11, 2019 12:01 am

AncientMemories wrote:
Sat Aug 10, 2019 10:33 pm
Should've actually gone to con, i guess :)

Also, the winner of the game is the one who gets the most solo's? Maybe there should be two winners, whoever gets the most solos and whoever has the highest average expected solo rate? Might be more complicated than it's worth, but for an extreme example, an extraordinarily lucky player who always puts russia as number one might win the game as I understand it if they happen to hit the 3% sweetspot an improbable number of times. Or do I misunderstand the scoring? Obviously that's unlikely, and the intent for the 1e9 games is to reduce the chances of something like that, but...
Yes, there is luck involved, but over the course of a million games where the average expected solo number will be somewhere around 50k (assuming around 20 players), the law of large numbers states that the likelihood of those two winners being unique is inconsequential. If two players submit the same strategy - thats when luck will determine the winner.
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Re: Game Theory Games

#29 Post by Claesar » Sun Aug 11, 2019 7:34 am

Squigs44 wrote:
Sun Aug 11, 2019 12:01 am
AncientMemories wrote:
Sat Aug 10, 2019 10:33 pm
Should've actually gone to con, i guess :)

Also, the winner of the game is the one who gets the most solo's? Maybe there should be two winners, whoever gets the most solos and whoever has the highest average expected solo rate? Might be more complicated than it's worth, but for an extreme example, an extraordinarily lucky player who always puts russia as number one might win the game as I understand it if they happen to hit the 3% sweetspot an improbable number of times. Or do I misunderstand the scoring? Obviously that's unlikely, and the intent for the 1e9 games is to reduce the chances of something like that, but...
Yes, there is luck involved, but over the course of a million games where the average expected solo number will be somewhere around 50k (assuming around 20 players), the law of large numbers states that the likelihood of those two winners being unique is inconsequential. If two players submit the same strategy - thats when luck will determine the winner.
Due to the large number of variables, I think we'll see large differences between players that submit the same sequence.

Now I'm going to estimate how many players submit the same order.

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Re: Game Theory Games

#30 Post by Claesar » Sun Aug 11, 2019 7:41 am

Wow, that was an easy calculation..

There's less than a 50% chance there will be a matching submission. I can't reveal my math as that would say too much about my thought process. I'll just send it to Squigs instead.

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Re: Game Theory Games

#31 Post by gimix » Sun Aug 11, 2019 9:21 pm

Happy on my shared second place - BobMcBob's fault :lol:
I think it would be very interesting to know how everybody chose where to build, and of course what would have Squigs44 choose, and why :-)

So i'm disclosing my line of thougth. I adopted a very simple heuristic: you only can be sure of getting points from the sc you build at, and have real hopes of points from the centres which are 1-step away - more remote centres have a high probability to be claimed by someone else. So for each sc i simply added its base value, and half the value of each sc's directly bordering with it. Under this assumptions Kie was a clear winner with a score of 73.5, and Bul was the second choice at 66.5. So i selected Kiel.

I had thought about working on more sophisticated versions, e.g. taking into account how many chances a bordering centre had of being contended, i.e. how many bordering centres it in turn had; but i didn't have time to work on it. The funny thing is, i strongly suspect the more sophisticated version would have led me to choose Bre - and BobMcBob would have won. So it seems i've been a sort of kingmaker, or my job was :-)

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Re: Game Theory Games

#32 Post by Squigs44 » Mon Aug 12, 2019 12:57 am

gimix wrote:
Sun Aug 11, 2019 9:21 pm
I think it would be very interesting to know how everybody chose where to build, and of course what would have Squigs44 choose, and why :-)
Even knowing where everyone chose, it wouldn't have mattered where I went, I could not have won. A Berlin might have been my choice though, or A Paris. When first planning it, I actually thought a fleet in either MAO, North Sea, or the Channel would have been a good play, but it was too crowded for that to have worked.
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Re: Game Theory Games

#33 Post by BobMcBob » Mon Aug 12, 2019 10:12 am

gimix wrote:
Sun Aug 11, 2019 9:21 pm
So i'm disclosing my line of thougth. I adopted a very simple heuristic: you only can be sure of getting points from the sc you build at, and have real hopes of points from the centres which are 1-step away - more remote centres have a high probability to be claimed by someone else. So for each sc i simply added its base value, and half the value of each sc's directly bordering with it. Under this assumptions Kie was a clear winner with a score of 73.5, and Bul was the second choice at 66.5. So i selected Kiel.
My reasoning was nowhere near that sophisticated. My reasoning went more like:
Rumania looks nice, it's pretty close to a lot of centers, and most of them are pretty high scoring. (Submit)
Actually, a lot of people are going to be attracted to that area for the same reason, let's choose something else.
Burgundy looks pretty neat, it's a move away from a bunch of centers, including the high-scoring Belgium. And then there's Brest a ton of centers two moves away as well, including Brest. (Edit)
The next day: Nope, far too many people are doin this for an off-center strategy like that to work, I need to be on a center to have a better chance. How about Kiel? That's next to Berlin, Denmark, 2 moves from Belgium, and Kiel itself is not bad points-wise either. Let's do that! (Resubmit)

Yeah, I didn't do an overly complicated analysis, it was very surface-level.

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Re: Game Theory Games

#34 Post by Squigs44 » Sat Aug 17, 2019 8:35 am

Hello,

Again, I am posting this late at night, because I was too excited about the results to wait. However, I do not have this weeks game quite ready, so that will have to wait for a little (hopefully I can get it up within 24 hours).

First off, lets start with some interesting tidbits, player insights sent to me via PM, and my own personal thoughts:

There were 16 participants. 10 submitted straight strategies, 2 submitted a mixed strategy between two options, and 4 submitted a mixed strategy between 3 options. I guess putting a limit on 5 strategies was pointless. The top 3 and bottom 4 players all used straight strategies, so using a mixed strategy seemed to be the "safe play" here.

I liked the fun words that were spelled by some entries, such as GIFT EAR or FRIGATE, or FIG EATR.

Claesar: "Anyone who doesn't put Russia last is a complete doofus." Most people agreed with Claesar, as all but one participant put Russia last in their strategies. Logically, putting Russia last is the optimal play in any scenario, since ANY other country would have a higher percentage. However, that one participant that put Russia last and was a "doofus" actually scored very well (but still didnt win).

90kicvesb: "I think there’s almost no chance that I’ll take first place in this contest. I predict that honor will fall to someone who goes with a non-mixed strategy and correctly picks which of (F,T,G) the rest of the field undervalues." You were correct. The winner of this game was able to correctly identify Turkey as the most undervalued country. Only 3.77 people picked Turkey first (2 straight strategies and 5 mixed strategies totaling 177%). The two who got first and second were the two with Turkey at the front of their straight strategy, and 4th, 5th, and 6th had Turkey in a mixed strategy. 3rd place was the only player to have Germany at the front of his straight strategy. France was definitely the overpicked country.

Claesar: "When you calculate various options and distributions though, if everyone ranks one F/T/G first in their sequence they still outscore anyone who puts Italy first. As a result, no one should. Therefore, the number of options becomes:

3*5*4*3*2*1*1=360

That number happens to be close enough to the 'birthday paradox' to know we'd need 24 sensible submissions to reach over a 50% chance to have a duplicate entry. From last round we can expect not to reach 24, so we can safely say the odds of a duplicate entry are less than 50%."

Well Claesar, you do remember last game when you picked A Bul with several others, right? It turns out that you just can't be original. You and RoganJosh both submitted FIEAGTR as straight strategies, the only instance of that happening this time around (although there was some overlap elsewhere if you count mixed strategies)

ItsHosuke submitted the following as his message: "Reverse psychology at its finest: FTGIEAR". While I applaud the boldness of this strategy, it didn't work so well. You actually got second to last. Last place goes to Kingdroid, the only person to put Italy as the first option. Italy's solo rate (12%) is lower than the average 1/7 (14.28%), so putting Italy first wasn't ever gonna work.

I did calculate out expected win rates (based on how many times you drew each country), and everyone's win rate was within at least 0.1% of their expected win rates, most of them within 0.02%. This deviance did shake up the standings a little bit, but the top two players were so far ahead that it didn't change the winner.

As far as guessing the solo rate goes, everyone overguessed! The closest was 90kicvesb with a guess of 17.3%. The actual value was 16.92%

And now for the results (Players, then win percentages):
1 KCelec 16.92
2 gimix 16.21
3 Temasek22 14.50
4 damo666 14.49
5 90kicvesb 14.43
6 tr1285 14.33
7 yavuzovic 14.31
8 teccles 14.28
9 BobMcBob 14.26
10 xorxes 13.97
11 Mercy 13.91
12 pyxxy 13.86
13 Claesar 13.91
14 RoganJosh 13.83
15 ItsHosuke 13.38
16 Kingdroid 11.98

KCelec had a strategy of TIEAFGR while gimix chose TGFIEAR. It turns out Italy as a second choice was a huge for KCelec, since he was able to secure Italy 54k games as opposed to gimix getting Germany in 32k of his. KCelec only had Russia in 24k games, while gimix has Russia in 148k of his games.

Congrats to KCelec, and I feel real bad for gimix who would have won last week if not for BobMcBob's entry and would have won this week if not for KCelec's entry. Can gimix continue this pattern into next week?

And finally, the best advice I received from this:
After stating that optimizing your country selection list would gain you only about 3% above average, 90kicvesb said "I recommend you work on your gunboat skills rather than trying to implement your country selection idea, Squigs" Good advice, I guess I will have to read up.
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Re: Game Theory Games

#35 Post by BobMcBob » Sat Aug 17, 2019 9:37 am

Dang. I realised I should have put Turkey first last night, after thinking it over again, but by then it was too late. Congratualtions gimix, you got me this time :)

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Re: Game Theory Games

#36 Post by damo666 » Sat Aug 17, 2019 9:53 am

At least it seems I won the mixed strategy race. Would be interested to know if anyone submitted either of my pair as straight and if so where they came. Thanks.

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Re: Game Theory Games

#37 Post by 90kicvesb » Sat Aug 17, 2019 3:54 pm

Thanks for quoting my submission, Squigs! I'll add my attempt to come up with optimal percentages for the first country selection.

I made the unrealistic assumption that everyone else would use the same weights as me. For example, if I pick (F) 60% of the time, then there’s a 0.4^6=0.4% chance that I could pick it up uncontested and a 6*0.6*0.4^5=3.7% chance that I could have a 50/50 with one other person. All in all, I have a 23.8% chance of getting assigned (F) given that I pick it with slot #1. If I decrease the weight of (F) to 50%, then the probability of getting assigned (F) given that I pick it increases to 28.3%.

Next, I played with the weights of slot #1=(F,T,G) until the overall solo rate from each was the same. Unsurprisingly, they were very close to 14.3% once balanced. The resulting weights were (50%,34%,16%).

I thought people might pick France too much, but made only a small bet, landing on a submission of:

45% FIEAGTR
37% TIEAGFR
18% GIEATFR

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Re: Game Theory Games

#38 Post by Claesar » Sat Aug 17, 2019 4:24 pm

90kicvesb wrote:
Sat Aug 17, 2019 3:54 pm
...
I thought people might pick France too much, but made only a small bet, landing on a submission of:

45% FIEAGTR
37% TIEAGFR
18% GIEATFR
Wow, we basically had the same thought process except that I estimated France would be underpicked. Hence my submission of FIEA followed by some irrelevant letters. I considered TIEA and GIEA as well, but ultimately went big.

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Re: Game Theory Games

#39 Post by Squigs44 » Sat Aug 17, 2019 8:04 pm

Correction: I erroneously ranked Claesar lower than he should have been. He should be above pyxxy obviously
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Re: Game Theory Games

#40 Post by pyxxy » Sat Aug 17, 2019 11:42 pm

Squigs44 wrote:
Sat Aug 17, 2019 8:04 pm
Correction: I erroneously ranked Claesar lower than he should have been. He should be above pyxxy obviously
I did appreciate not being unlucky 13th :razz:

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