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Subject: First-player advantage (with stats from Boiteajeux) and how to fix it rss

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Jake Talley
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THE STORY

After playing Trajan again on Tuesday and mentioning while explaining it that there was a small first-player advantage, I started wondering exactly how big this might be and whether it could be easily fixed, as komi are used in Go. So, I got in touch with Diplojak at Boiteajeux.net and he was nice enough to provide the final scores of all the Trajan games that have been played on the site. This was a total of 13749 games. I chose to remove all the games where the maximum score was less than 50 (of which there were 274), assuming that these were abandoned. I probably could have raised this threshold a bit, but it doesn't affect the results much at all.

Of the remaining 13475 games there were 9702 with 2 players, 1980 with 3 players, and 1793 with 4 players, which I think is a large enough sample to lend a bit of weight to the findings. I didn't bother trying to calculate confidence estimates for anything. What exactly were these findings?

THE RESULTS

As suspected, being earlier in turn order confers a small advantage. For 2-player games the mean, median, and standard deviation of first player scores were 131.8375, 133, and 26.3895, respectively. For second player scores they were 128.1322, 128, and 26. Thus the first player enjoyed a mean advantage of about 3.7052 points, which resulted in the first player winning 53.68% of games. If the second player is awarded 3.7 points to begin the game then the first player's win share becomes 50.43%. After discussing 3- and 4-player games I'll give an easy way to implement this option.

For 3-player games the mean, median, and standard deviation of first player scores were 116.9338, 116, and 23.3811. For second player scores they were 114.1283, 113, and 22.7058. For third player scored they were 111.5773, 111, and 23.2495. This results in a first-player advantage of 2.8056 points over the second player and 5.3566 points over the third player. Win shares for first, second and third players were 37.32%, 34.14%, and 28.54%. After adding the offsets to the scores these change to 33.69%, 33.79%, and 32.53%.

For 4-player games the mean, median, and standard deviation of first player scores were 107.2967, 106, and 23.5988. For second player scores they were 105.3475, 106, and 22.8469. For third player scores they were 103.5945, 104, and 22.5013. For fourth player scores they were 101.966, 102, and 22.8121. The resulting first-player advantage was 1.9492 points over the second player, 3.7022 points over the third player, and 5.3307 points over the fourth player. Win shares for 1st-4th were 28.33%, 25.82%, 24.09%, and 21.75%. After adding the offsets these change to 24.76%, 24.65%, 25.43%, and 25.15%.

I think it should be fairly clear that the first player has a small, but still significant, advantage in Trajan. For a 4-player game the first player wins about 30% more often than the fourth player, which is unacceptable to me. Luckily, it's pretty simple to correct this so that every position is within half a percent of even.

TRAJAN KOMI

For a 2-player game, the second player starts with 3 points.
For a 3-player game, the second and third players start with 3 and 5 points, respectively.
For a 4-player game, the second, third, and fourth players start with 2, 4, and 5 points, respectively.

In all cases ties are broken in turn order, but starting with the last player. In a 2-player game, the second player wins ties. In a 3-player game, the third player wins ties, followed by the first player. In a 4-player game the tiebreak order is fourth, first, second.

OTHER INTERESTING BITS

The maximum scores for 2p, 3p, and 4p games were 246, 207, and 203 points, respectively. The highest scores which *lost* were 190 (this was a 191-190 game!), 168, and 159 points. If anybody thinks of other things which might be neat to know just send me a GM.

Hopefully some of you find this interesting, and maybe it will help balance your future Trajan games. If a lot of people find it interesting, I might try and extend this to other games that have popular online implementations. Thanks again to Diplojak for providing the data for me to work with.
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Shawn Fox
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The biggest issue is the fact that the game ends as soon as the marker crosses the starting position. I always play so that all players get the same number of turns in face to face games.
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Jonathan Er
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i have detailed my own fix'it for the game at my podcast website

do check it out !

http://www.pushurluckpodcast.com/2013/07/fixit-for-dummies-t...





Jon
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Jake Talley
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sfox wrote:
The biggest issue is the fact that the game ends as soon as the marker crosses the starting position. I always play so that all players get the same number of turns in face to face games.


sure, but in this game i don't think equal turns is a good solution. a lot of thought has to go into how your choice of tray to empty affects the time track, and if you know you're guaranteed another turn this takes away some of that tension. it's not exactly in the spirit of the design.

Jion wrote:
i have detailed my own fix'it for the game at my podcast website


i also wouldn't be comfortable suggesting additional starting resources since that subtly suggests a strategic direction for each player position and i have no good sense of the effective value of the resource options. if players are simply given points then the design and balance of the rest of the game is completely unaffected.
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Maarten D. de Jong
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I ran Student's t-tests on the given numbers (which is what really decides if matters are significant or not) and came up with the following for the 2P-case:

— H0 is rejected at P < 0.0001, therefore the difference is extremely significant at 0.05-level;
— with the correction, H0 is accepted at P ~ 0.061, therefore the difference is no longer significant at 0.05-level.

In short: in the 2P case the difference is real, and the correction works. I then ran a t-test on the difference between 1st and 2nd player in the 3P-case, and concluded that although less extreme, H0 is still rejected at P ~ 0.001. This is due to the much smaller sample size... But since the raw difference in the means is still comparable, and that the sample size in the 4P-case is comparable to that of the 3P-case, there's need for another calculation here, the difference is significant.

I will adopt these corrections as a sort of komi. What surprises me most is their smallness given the amount of points one expect to obtain: it's really just a few points. By the way, I'm also not in favour of giving everyone the same amount of actions, and not just because as was already said that it it's not something which comes easily to the design: the numbers presented indicate quite clearly that the value of an action depends on where you sit in in the player sequence, and with how many people you play. (I suspect this is because of the timing of determining the Voice of the People, which is subject to unequal amounts of player actions too.) You'd have to introduce fractional actions to make it all work again: points are simply easier.

That all said: because H0 is accepted at P ~ 0.061 in the corrected 2P-case, I could calculate that, assuming that the mean and sd don't change in the next 10.000 games (which isn't entirely correct), the difference becomes significant again after only +/- 800 more games have been played (but only weakly so, of course, it's simply looking when the crossover to P = 0.05 happens). So the imbalance remains. The question is whether you worry about an imbalance which shows up only after 10.500 games have been played. But I believe the OP already sufficiently addressed this by changing the tiebraker rules.

Good job, have a thumbsup on me.
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Bryan Thunkd
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Having played mostly two player games, my final scores are rarely as close as 3 points. That seems like a fairly small swing given the final scores.
 
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cross reference:

Let's brainstorm a solution to the first player imbalance, for those that think there is one

Imbalance due to different number of actions?
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Neil Christiansen
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I really like Jion's second fix from his blog.

^^^^^^^^^^^^
New Rule:

Start player starts as per basic game setup. Subsequent players get additional benefits as follows:
2nd Player – One additional resource / shipping card ( randomly drawn from top of deck )
3rd Player – One additional worker in the military camp
4th Player – One additional worker in the construction camp
(Benefits do not stack ! 4th Player does not get one card + one worker in both camps )
^^^^^^^^^^^^^

Marteen, I am not sure one should ever "accept" H0 in statistics, but it may be semantics. I might fail to rule out sampling error at a certain level of confidence, but the difference in means is the difference in means. They are not equal.
 
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scud80 wrote:
sfox wrote:
The biggest issue is the fact that the game ends as soon as the marker crosses the starting position. I always play so that all players get the same number of turns in face to face games.


sure, but in this game i don't think equal turns is a good solution.


But the point is that this is a significant flaw in your analysis. If player 1 wins by three points, how do you know the victory isn't caused by player 1 getting an extra action? Most actions are worth roughly three points (as explained here). If we assume that player 2 was denied one more action than player one, simply giving them that action covers your calculated discrepancy. (It also provides a solution that is dependent on the skill of the player, which I would prefer over a flat handicap.) With your solution, if you give player 2 a specific point handicap, that player now has an advantage in games where both players have an equal amount of turns. What we really need to see here is the point spread of games where each player has an equal amount of turns versus games when player one ends the game. My guess is that the website you culled your data from doesn't provide this information.

As an aside, does your data includes the number of turns per game? I am curious to see how close the actual points per action value is to my estimated 3.25.
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Shawn Fox
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There are lies, damn lies, and statistics. At an aggregate level your analysis appears to be correct, but if you look at it in detail you'll find that it fails. Try rerunning it (if possible) based on which player got the last action. What you'll find is that by giving players extra points at the start of the game you have given large advantages to the 3rd/4th players if they manage to be the last player to take an action in the game.

Basically it is a duct tape solution which does the opposite of what you intend and hands a big, fairly random, advantage to the 3rd/4th players instead of the advantage going to the first player. Since each action is worth around 7 points on average, the only way to balance the game is to insure each player gets the same number of actions.
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Neil Christiansen
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I think his correction does not hand "a fairly big, random advantage" to anyone. It adds a small, systematic advantage. It is based on averages, and averages are pretty wonderful things when based on thousands of trials.

Remember, the players earlier in turn order can end the game early fairly easily. I have seen it done countless times. As start player, taking a suboptimal move that ends the game is often better than taking my optimal move and allowing others to as well.

Many games do this by sliding starting resources.
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Shawn Fox
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chris1nd wrote:
I think his correction does not hand "a fairly big, random advantage" to anyone. It adds a small, systematic advantage. It is based on averages, and averages are pretty wonderful things when based on thousands of trials.

Remember, the players earlier in turn order can end the game early fairly easily. I have seen it done countless times. As start player, taking a suboptimal move that ends the game is often better than taking my optimal move and allowing others to as well.

Many games do this by sliding starting resources.


The players early in the turn order have no more capability for ending the game on their turn than anyone else does.
 
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Mathue Faulkner
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I don't like the 'equal turns' solution or the 'resource' solution. This is the only one that may see my table...
 
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Neil Christiansen
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But later players can only end the game having an equal number of turns as earlier players. Earlier players can end the game having an extra turn over later players. What am I missing here?

There clearly is an advantage. For example, the odds of flipping a coin 9,000 times and getting that many heads as the first player wins in a 2-player game is astronomical. Ditto for same logic for 3-player where first player wins 37% or more.

The question is what to do about it.

We lay out k+1 bonus tiles at start of game and choose in reverse turn order to try to reduce the advantage. We then do card selection same.
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Maarten D. de Jong
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chris1nd wrote:
Marteen, I am not sure one should ever "accept" H0 in statistics, but it may be semantics. I might fail to rule out sampling error at a certain level of confidence, but the difference in means is the difference in means. They are not equal.

Damn skippy it's semantics. 'Accepting H0' is standard 101-level statistical jargon; are you sure you want to overturn this convention given that everyone on Earth who learned about H0 and H1 uses this phrase?

sfox wrote:
The players early in the turn order have no more capability for ending the game on their turn than anyone else does.

Do you mind to speculate as to why we shouldn't fix the unequal number of turns imbalance when each of the four years is ended then? Relatively speaking the loss of a single action bites much harder here than it does over the course of the entire game, and Feld knows what the effect of varying amounts of 'time points' per year is (since the track counter is not reset to the starting position).
 
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I don't love the idea of handing out points to start. What if, after the quarter ends, turn order is shuffled? In a 4P game, each player would get one chance to start the next quarter. In a 2P game, two chances. In a 3P game, one player gets an extra "first player" chance. This maintains the tension present from not knowing when the quarter is over, and also adds some potential tactical plays in anticipation. I know it's not perfect, but perhaps this would lessen the advantage enough to be negligible.
 
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cymric wrote:

Do you mind to speculate as to why we shouldn't fix the unequal number of turns imbalance when each of the four years is ended then? Relatively speaking the loss of a single action bites much harder here than it does over the course of the entire game, and Feld knows what the effect of varying amounts of 'time points' per year is (since the track counter is not reset to the starting position).


That is a trivial one to answer. It is an advantage to go early in the next round. It is also very likely that if you get 1 less turn in the prior quarter you are more likely to get 1 extra turn in the next quarter, so it balances out nicely.

At the end of the game, however, there is no next quarter to make up the difference, thus you just end up with -1 turn over the course of the game and just flat out lose the point value of the turn (approximately 7 points if you know what you are doing).
 
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sfox wrote:
That is a trivial one to answer. It is an advantage to go early in the next round. It is also very likely that if you get 1 less turn in the prior quarter you are more likely to get 1 extra turn in the next quarter, so it balances out nicely.

No, it's not trivial, otherwise I wouldn't have asked. You now state that it all balances out, yet overall it obviously doesn't. One thing has to give: what's it going to be? Either it's all imbalanced, or none is. Your argument is that it is all imbalanced, so the years have to be imbalanced too. And that is a much more serious ('point swingy') issue in the sense that it introduces larger discrete point differences than can be accounted for by a simple '1 action equals x points'-scheme.
 
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Ya know I have always found these discussions quite interesting. When I was a kid I spent a lot of time "fixing" my games too. When I started woodworking seriously I was designing things that were remarkably complex (as I came to realize when building them). There is an elegance in the idea "less is more," there is an elegance in KISS (Keep It Simple Stupid).

There is nothing done to balance sports teams and clearly there are teams which have advantages over others. The only balance is often more than one game over the season. The challenge and consequently the excitment is shutting down those advantages.

I fear spending too much time "fixing" and advantage will lead to utterly sterile games.

Let's say there is a distinct advantage for the start player in a 2p game. Instead of futzing about with the game why not just figure the winner over the combined score of two games with each player getting to go first.

The games of today are rather complex systems (compared to previous games), so one tweak here can have results utterly unexpected.

Likewise this concern over equal game turns makes me scratch my head as well. Games like this only provide different opportunities for for bringing about the end game. When the advantage is yours you should be striving to end the game when it isn't you should be figuring out how to optimize the time you have left to shift that advantage.

If we are to get so concerned about everything being equal instead of playing any of these complex games we should just sit around and flip a perfectly balanced coin, heads I win, tails you lose (my favorite rules for this one, by the way).
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cymric wrote:
sfox wrote:
That is a trivial one to answer. It is an advantage to go early in the next round. It is also very likely that if you get 1 less turn in the prior quarter you are more likely to get 1 extra turn in the next quarter, so it balances out nicely.

No, it's not trivial, otherwise I wouldn't have asked. You now state that it all balances out, yet overall it obviously doesn't. One thing has to give: what's it going to be? Either it's all imbalanced, or none is. Your argument is that it is all imbalanced, so the years have to be imbalanced too. And that is a much more serious ('point swingy') issue in the sense that it introduces larger discrete point differences than can be accounted for by a simple '1 action equals x points'-scheme.


Did you miss the 2nd paragraph in my reply? Why did you exclude it when you quoted my reply? As I said, when the round ends on your turn you gain an advantage in that round but you give up something in the next round by going last in that round. Thus there is no net gain unless it is the last round, in which case there is no next round to allow the other players to make up the difference of the missing turn. How is my reply not clear on that?

[edit] Also even with an even number of turns, I do agree there is still a first player advantage, but it is small enough to be acceptable to me. Without playing with a house rule of some sort, Trajan currently has a huge first player advantage that is unacceptable.
 
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Maarten D. de Jong
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sfox wrote:
Did you miss the 2nd paragraph in my reply? Why did you exclude it when you quoted my reply? As I said, when the round ends on your turn you gain an advantage in that round but you give up something in the next round by going last in that round. Thus there is no net gain unless it is the last round, in which case there is no next round to allow the other players to make up the difference of the missing turn. How is my reply not clear on that?

*Sighs*. I would have assumed that you yourself realised that in the very first year more time track points are used compared to later years because the time track counter is not reset to zero. You missed that detail in my original question, didn't you? Shall we dispense with the annoying 'you didn't read what I wrote'-thing and continue to discuss like rational adults? Thank you.
 
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An interesting thread, but I wonder if we can adopt a market-based approach. The Princes of Florence is deigned to have similar conditions where turn order matters to winning the game (second seat is deemed to be best). During the World Board Gaming Championships, seat position is auctioned prior to game start in half-point VP margins. These basically put you in the hole prior to game play. So, my questions are ...

* Is Trajan played at WBC?
* Do they auction seat position?
* What do players "pay" for start player?
* Is it worth what they paid, i.e. did they win in spite of the payment?

The reason I like this approach is I'm assuming folks who make it to WBC to play Trajan are skilled at the game and are aware of the start player advantage.
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cymric wrote:
sfox wrote:
Did you miss the 2nd paragraph in my reply? Why did you exclude it when you quoted my reply? As I said, when the round ends on your turn you gain an advantage in that round but you give up something in the next round by going last in that round. Thus there is no net gain unless it is the last round, in which case there is no next round to allow the other players to make up the difference of the missing turn. How is my reply not clear on that?

*Sighs*. I would have assumed that you yourself realised that in the very first year more time track points are used compared to later years because the time track counter is not reset to zero. You missed that detail in my original question, didn't you? Shall we dispense with the annoying 'you didn't read what I wrote'-thing and continue to discuss like rational adults? Thank you.


I didn't miss any details, I just don't see why you think that has any application to your argument. If anything it reinforces my argument. It becomes even more likely that if a player got +1 action on round X they will get -1 action on round X+1 and thus will balance out. Modifying the rules such that all players get the same number of turns by the end of the game also guarantees that in the worst case the last players will get +1 action in the last round if they did not do so in an earlier round.
 
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I am pretty sure that one can only either reject or fail to reject the null. One never accepts it, as the null cannot be proven. Do you accept something that cannot be proven? Certainly, Fisher never used that language to my knowledge. Neyman and Pearson changed it up a bit, but the point is essentially the same.

Why I am arguing this when I think statistical tests of the null are a fool's errand anyway, I dunno. The sample means are the means. They are our best guess of the population means. And the sample means are not equal.
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Thanks for doing this. Very interesting.

Any chance of sharing the raw data, so others can futz around with it?

 
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