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Subject: Player Count and Scores rss

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Dan Likos
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New to the game, but have now played a few 5 player games and most recently, one 4 player game. Our scores went up, and we were trying to figure out if it was because we were getting better, or if scores increase as player count decreases.

Anyone chime in on this? Do scores increase as player counts decrease?
We noticed that there were more tiles left in the sack when the game completed.

Thanks.
 
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Alex Bove
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You should generally expect scores to increase as player counts decrease. This is especially true in a 3-player game, since each player has 4 possible bids each epoch.

With the same total number of positive point tiles, and fewer players competing for them, it stands to reason that scores would tend to go up, right?
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Dan Likos
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montu wrote:
You should generally expect scores to increase as player counts decrease. This is especially true in a 3-player game, since each player has 4 possible bids each epoch.

With the same total number of positive point tiles, and fewer players competing for them, it stands to reason that scores would tend to go up, right?


Logic is definitely sound for seeing an increase in the 3 player game as there are more suns available.
 
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Christopher Dearlove
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dlikos wrote:
montu wrote:
You should generally expect scores to increase as player counts decrease. This is especially true in a 3-player game, since each player has 4 possible bids each epoch.

With the same total number of positive point tiles, and fewer players competing for them, it stands to reason that scores would tend to go up, right?


Logic is definitely sound for seeing an increase in the 3 player game as there are more suns available.


That's not the best logic.

The number of suns is fixed, but you need fewer to end the game. That is roughly set up to give a similar number of tiles per player. Which would indicate similar scores. Except that with fewer players they can be more efficiently distributed, resulting in larger scores. But not massively larger.

(A more accurate assessment of the rules per player might also indicate that the number per player isn't exactly balanced.)
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Alex Bove
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Dearlove wrote:

That's not the best logic.

The number of suns is fixed, but you need fewer to end the game. That is roughly set up to give a similar number of tiles per player. Which would indicate similar scores. Except that with fewer players they can be more efficiently distributed, resulting in larger scores. But not massively larger.


Actually, I think my logic is just fine. Let's look at it more carefully.

The number of total non-Ra, non-disaster tiles in the game is always 140 (40x of monuments; 25x of civs, pharoahs, and niles; 12x of floods, 8x of gods, 5x of gold), no matter the player count. The number of total Ra tiles is also the same (30) in each player count, though the odds of ending the game before all the non-Ra tiles are slightly increased as the player count decreases.

Let's assume a totally even distribution of tiles is drawn, and let's assume that each player gets a roughly equal share of those tiles throughout the game. How does that look for each player count?

With 5 players, there are 45 total suns/bids in the game, 15 bids per epoch. We should expect 36-37 non-disaster tiles to be drawn for every 10 Ra tiles (3.65 non-Ra tiles for every Ra). In a 5-player game, then, each player should expect to win about 2.5 tiles per bid, if all bids are used and an average number of total tiles are won.

With 4 players, there are 36 total suns/bids in the game, 12 bids per epoch. Let's say that the Ra tiles come out at exactly the same ratio per non-Ra tile as in a 5-player game. So at 3.65 non-Ra tiles for every Ra, we should expect 32-33 non-disaster tiles (3.65x9) to be drawn every round. Now each player should expect to win about 2.75 tiles per bid.

With 3 players, there are also 36 total suns/bids in the game, 12 bids per epoch. Again, using the metric of 3.65 non-Ra tiles for every Ra, we should expect about 29 (3.65x8) non-disaster tiles to be drawn every round. That works out to 2.4 tiles per bid, but each player gets an extra bid!

This means that the total numbers of tiles each player can expect to win in each game is as follows:

5-players: 22.5
4-players: 24.75 (+10% vs. 5-player)
3-players: 28.8 (+16% vs. 4-player)

We can clearly see that as the player count decreases, the number of tiles players can expect to win in a game increases. Thus, we should expect scores to increase as the player count decreases.

Note: I've removed disasters from this equation because the effect of disasters is impossible to know. Even if we calculated that disasters would occur at even intervals throughout the game (so 10 disasters in 180 total tiles, or 1 disaster per 18 tiles, or about 3 per epoch), we couldn't ever calculate the impact of disasters on scores or tile counts because disasters have an asymmetrical effect on players.

Assuming everyone only takes disaster tiles when they have no effect (i.e. they never cause the player to have to discard any tiles), then the total effect of disaster tiles is that they are blanks. This has a minimal effect across player counts, as in a 5-player game, players should expect 2 blank tiles each game (11.25% of their total tiles) while in a 4-player game, they should expect 2.5 blanks per game (9.8%) and in a 3-player game their EV is 3.33 blanks per game (8.72%). These slight increases in ineffective tiles do not offset the overall increases in helpful tiles that we see as player numbers decrease, as illustrated above.


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