| 1 |
[quote]This literally gives you a 1/3 chance to lose against two Null AIs![/quote]
|
1 |
[quote]This literally gives you a 1/3 chance to lose against two Null AIs![/quote]
|
| 2 |
That's a very good observation and quite solid proof that the proposed formula doesn't work.
|
2 |
That's a very good observation and quite solid proof that the proposed formula doesn't work.
|
| 3 |
\n
|
3 |
\n
|
| 4 |
The main problem is that any team balancing algorithm needs to make assumptions about how team composition affects team strength/elo. This is basically the choice of f and g in the OP, although it would be easier to define it as a function p(elo vector of team) that produces team strength/elo, which is essentially a norm, like the p-norm @Brackman mentioned (see again the OP, or [url=https://en.wikipedia.org/wiki/Lp_space#The_p-norm_in_finite_dimensions]glimpse over this[/url] <- just look at pictures and formulas).
|
4 |
The main problem is that any team balancing algorithm needs to make assumptions about how team composition affects team strength/elo. This is basically the choice of f and g in the OP, although it would be easier to define it as a function p(elo vector of team) that produces team strength/elo, which is essentially a norm, like the p-norm @Brackman mentioned (see again the OP, or [url=https://en.wikipedia.org/wiki/Lp_space#The_p-norm_in_finite_dimensions]glimpse over this[/url] <- just look at pictures and formulas).
|
| 5 |
\n
|
5 |
\n
|
| 6 |
Your example with the teamstrength change looks correct (although you did mess up the signs for the expected outcomes), but the question still stands how to map/combine individual strength/elo to team strength/elo.
|
6 |
Your example with the teamstrength change looks correct (although you did mess up the signs for the expected outcomes), but the question still stands how to map/combine individual strength/elo to team strength/elo.
|
| 7 |
\n
|
7 |
\n
|
| 8 |
Some observations:
|
8 |
Some observations:
|
| 9 |
- Team strength (as used by you) must be in the interval (0; ∞]. It cannot be < 0 (negative probability?), but it can be arbitrarily large, because for every team with strength p1, one can imagine a stronger team that will beat the first one 2 out of 3 times and thus must have a strength p2 = 2*p1. It could theoretically be zero, but not in practical applications.
|
9 |
- Team strength (as used by you) must be in the interval (0; ∞]. It cannot be < 0 (negative probability?), but it can be arbitrarily large, because for every team with strength p1, one can imagine a stronger team that will beat the first one 2 out of 3 times and thus must have a strength p2 = 2*p1. It could theoretically be zero, but not in practical applications.
|
| 10 |
\n
|
10 |
\n
|
| 11 |
- According to @Brackman p needs to (essentially) satisfy the triangle inequality to fit the known data: Adding a "variation" elo vector (like [-300; 100; 200]) to a team of equally strong players (like [1500, 1500, 1500]) increases their strength (so p([1200; 1600; 1700]) > p([1500; 1500; 1500])). (Note that this is not inherent to the math used! It would be possible that serious anti-synergies in teams with high elo variance would cause the opposite).
|
11 |
- According to @Brackman p needs to (essentially) satisfy the triangle inequality to fit the known data: Adding a "variation" elo vector (like [-300; 100; 200]) to a team of equally strong players (like [1500, 1500, 1500]) increases their strength (so p([1200; 1600; 1700]) > p([1500; 1500; 1500])). (Note that this is not inherent to the math used! It would be possible that serious anti-synergies in teams with high elo variance would cause the opposite).
|
| 12 |
\n
|
12 |
\n
|
| 13 |
- If all elo values increased by the same flat amount, the ratios of the team strengths would need to stay the same. This pretty much implies an exponential approach (which conveniently also satisfies the first point).
|
13 |
- If all elo values increased by the same flat amount, the ratios of the team strengths would need to stay the same. This pretty much implies an exponential approach (which conveniently also satisfies the first point).
|
| 14 |
\n
|
14 |
\n
|
| 15 |
- It is still unclear how different team sizes (= vector lengths) should affect p. Is one player with 5 coms stronger than 5 players with one com each (all of equal elo)? I reckon this largely depends on the players, and it's conceivable that for high numbers of extra coms, multiple players get an edge through micro advantage, but it's hard to judge how the force concentration (and increased coordination) balance that out in the general case.
|
15 |
- It is still unclear how different team sizes (= vector lengths) should affect p. Is one player with 5 coms stronger than 5 players with one com each (all of equal elo)? I reckon this largely depends on the players, and it's conceivable that for high numbers of extra coms, multiple players get an edge through micro advantage, but it's hard to judge how the force concentration (and increased coordination) balance that out in the general case.
|
|
|
16 |
\n
|
|
|
17 |
- All of this has many parallels to the 1v1 probabilities, because that's essentially just a special case of the above.
|
math bork 2nd try