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Topic: Google AI Contest Galactic Conquest (Read 22539 times) |
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Fritzlein
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Re: Google AI Contest Galactic Conquest
« Reply #30 on: Sep 15th, 2010, 2:13pm » |
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on Sep 15th, 2010, 8:31am, jdb wrote:So I need a method/formula for deciding which neutral planets to capture.
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One possible ranking is the total gain in troops the planet will produce in the remainder of the game. Currently the limit is 200 turns, but any large number will work. This gives the following formula:
Fitness = (200 - Distance) * GrowthRate - Troops |
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Games never last 200 turns, so a formula looking that far ahead is bound to be weak. Long-term payoff will be weighted far too much relative to short-term sacrifice.
My inclination is the complete opposite, namely to make the fitness function entirely which planet pays off first.
PaybackTime = Distance + Troops/GrowthRate
Whichever has the smallest PaybackTime is the best.
Planet,Troops,Growth,Distance,PaybackTime
A,8,5,6,7.6
D,31,5,4,10.2
E,30,5,7,13
C,4,4,14,15
B,67,5,8,21.4
Planets A and D are best to take. Note that even if we feel compelled to delay one turn to take D because we are concerned about home defense, D still pays back sooner than E.
If Sconibulus' method of calculating planet safety is not too complicated to incorporate, that would be excellent in that it allows slightly more aggression. Insuring home safety might still be slightly more conservative than optimal, but barely so.
For example, some bots are coded to empty out their home world on turn one to take as many neutrals as possible. Against such an opponent, we could send 65 ships on turn two to take their home world, a far better investment than attacking any neutral planet.
We could get marginally dinged if we attack two neutrals while they attack three. Then they will still have enough for home defense and be ahead in the colonization race. However, on turn two we would sense the safety of our home world and launch against a third and fourth neutral planet, only one turn behind them.
Optimal play is probably some sort of rock-scissors-paper situation where have some probability of not launching any colonization on turn one, just to keep them honest by having enough in reserve to attack their home world on turn two if they deplete it. However, I wouldn't really worry about optimal play at this point. Being one turn behind on colonizing one world when the opponent anticipates exactly what we will do is likely to be dominated (at the current level of bots) by not making the mistake of losing our homeworld and capitalizing on some opponents making the mistake of losing their homeworld. I have become convinced that it is most practical to leave the game theory of simultaneous moves out of it for the present.
Having given a fairly simple fitness function, let me complicate it by suggesting that planets nearer the center of the board are better to take than planets on the periphery. We generally don't want to attack neutrals that are closer to the enemy than to us. However, if we colonize toward the center while the enemy colonizes away from the center, that brings more neutrals within our range while taking neutrals out of his range. The planets we control are safer, i.e. he is farther from attacking us, so our behind-the-lines ships are almost all free to move about. The planets he controls are less safe, because we have a central base for attacking them, so he is less free to move about.
The strategic advantage of having more troops in the center is worth some sacrifice of not having the larger and/or faster-growing army. If the opponent's strength/growth is all on the edge, we will often be able to take central planets from him before his reinforcements arrive. I'm not sure what the proper balance between the two is, but I am confident it is important. It will be like chess: when you have centralized pieces, good things will happen later on, even if you can't see exactly how it will unfold at the time to you are moving pieces toward the middle.
The second complication I want to propose is that my PaybackTime is only a relative measure. There also needs to be some sliding scale whereby we don't take even the neutral planet with the best PaybackTime, because our ships are better spent fighting the enemy than fighting neutrals. There needs to be a factor that indicates the urgency of fighting the enemy in the center. The hotter the central battle, the more we just send all troops there and ignore colonization entirely. I suspect most games will see a steady increase in this "hotness" factor, starting very cool when home worlds are far away, when colonization is everything, and gradually heating up to the point that fighting the enemy is everything.
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Fritzlein
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Re: Google AI Contest Galactic Conquest
« Reply #31 on: Sep 15th, 2010, 3:04pm » |
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on Sep 15th, 2010, 1:43pm, Hippo wrote:But I am not sure if sending to A, to D and rest to the opponent is not even better  . |
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Your strategy (9 to A, 32 to D, 59 to enemy home) is exceedingly reckless. Suppose the opponent thinks, "I will wait one turn to see if my opponent is a complete maniac. Sure, I will lose a turn, but if he makes a bad move, I can crush him."
Then on turn two your opponent has 105 ships to deploy. He sees what you have done. He knows that if he attacks your home planet, the maximum defense you can muster will be 45 from home planet growth, plus 6 returning from planet D. An attack of 52 beats you. For his home defense he will have 35 in growth, thus needing only 24 to stave off your attack. 105-52-24 = 28 ships left to deploy, with two great options:
A) He can just rely on his home world and yours, send the extra ships to your home world. By turn nine he will have equal to your growth rate and more ships than you do, because he invested nothing attacking neutrals.
B) He can take his own planet A and C with the ships left over from taking your home planet and securing his own. That will leave him with approximately the same ships you have and a much higher growth rate.
So, to summarize, the opponent can beat your "better" move by the cunning strategy of doing nothing whatsoever on turn one. But he doesn't have to be that cunning. If he only makes the opening move suggested by Sconibulus, he will not only be equal with you on colonization speed, he can calculate on turn two that he has 40 free ships not needed for home defense. That's not enough to counter-attack your home world, but it is enough to launch a fleet to colonize his planet E. Your ships are all in the air, so you can't colonize anything else. The eventual fight on his home planet costs you each equal ships, but in the mean time, he is winning the colonization race.
Thus an immediate attack on the opposing home world, if it doesn't win, is a terrible move.
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Fritzlein
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Re: Google AI Contest Galactic Conquest
« Reply #32 on: Sep 15th, 2010, 3:31pm » |
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on Sep 15th, 2010, 2:13pm, Fritzlein wrote:| The strategic advantage of having more troops in the center is worth some sacrifice of not having the larger and/or faster-growing army. If the opponent's strength/growth is all on the edge, we will often be able to take central planets from him before his reinforcements arrive. |
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To expound on this point, I notice that you didn't list Planet F as another colonization option:
Planet,Troops,Growth,Distance,PaybackTime
A,8,5,6,7.6
D,31,5,4,10.2
E,30,5,7,13
C,4,4,14,15
F,29,4,9,16.25
B,67,5,8,21.4
You didn't list it because it is only distance 10 from the opponent, so even if we took it, the depletion in forces from fighting the neutrals would leave us vulnerable to being retaken by the opponent, and not ever getting the payback from our initial investment of ships.
Now, let's look what happens if our first turn is less aggressive than the opponent's first turn, thus leaving more of our ships centralized. If we end up with 25 more ships on our home planet than the opponent has on his, Planet F is suddenly no longer too risky to take. We can take it and hold it. Since we are one turn closer, we can wait to see any counter-attack the opponent launches, and neutralize it by launching reinforcements one turn after.
Unfortunately, for this particular map, having a cautious opening with centralized troops won't pay off unless the opponent makes a big error. If, for example, we use turn one to take planets AD, while he uses turn one to take planets ADE, on turn two we will note that we can take either planet E or F, and decide that E is better, merely ending up one turn behind.
You only have to jigger the numbers slightly, however, (e.g. swap their growth rates of 4 and 5) to make planet F the better choice. In that case waiting one turn to watch the opponent incorrectly take planet E will pay off substantially when we take planet F on turn two. My hunch is that the small losses of being a turn behind are less important than the big gains of often getting a better planet.
Also waiting one turn will leave the troop strength to punish any opponent (and there will be many) who incorrectly attacks planets ACDEF on the first turn.
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Sconibulus
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Re: Google AI Contest Galactic Conquest
« Reply #33 on: Sep 15th, 2010, 3:34pm » |
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I don't think Payback time is quite as important as growth rate, although it is an important metric, and definitely should be used as, at minimum, a tiebreaker.
Perhaps a better fitness function for purely safe planets would be 5*GrowthRate-3*PaybackTime, although that would almost certainly need to be empirically optimized. I think looking solely at payback time would lead to a shortsighted program that potentially grabs a lot of cheap small planets passing over those that require a more substantial investment, but pay off more ships over time.
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Hippo
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Re: Google AI Contest Galactic Conquest
« Reply #34 on: Sep 15th, 2010, 4:02pm » |
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on Sep 15th, 2010, 3:04pm, Fritzlein wrote:| Thus an immediate attack on the opposing home world, if it doesn't win, is a terrible move. |
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OK this has something to do with the flexibility cost .
The variants I have written are all meant as possible candidates each achieving some goal. None of them is best, each has countarplays.
Without risk of charging to your opponent attacking as much planets as possible/safe is the best option. Sending most of the troops to opponent is risky move leading to either fast loss or fast win.
I agree with the stone-paper-scissors and this is how I tought about mentioned variants A)....
One must probably run several scenarios to find whose variants are fully dominated by others and in the remaining "strongly connected component" chose ratios to chose particular of them. The variants which allow good play against most of them had to gain higher probability.
I agree more flexibility allows better play in the midgame and centralised trops are more valuable.
But the grow ratio remains the main principle ... sending grow ratio ships to your nearest opponent planet (or neutral planets the total number of send ships exceeds their defense and your ships would fight opponent's) seems increases your lead
(as it reduces opponent chances to concentrate on attacking your planets) Of you can be faster than opponent growing in some region and slower in another. In that case you have to send troops to rebalance these regions.
One more though probably hardly to made usable ... having almost symmetrical position with say one small planet more so you have slightly higher grow ratio ... in that case you could try to rearange your troops almost symmetrically to your opponent to mimic his moves ... but being one turn late could be easily fooled.
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| « Last Edit: Sep 15th, 2010, 4:09pm by Hippo » |
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jdb
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Re: Google AI Contest Galactic Conquest
« Reply #35 on: Sep 15th, 2010, 4:08pm » |
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OMG, all this analysis for just the opening move! I'll never get the program finished.
Quote:| PaybackTime = Distance + Troops/GrowthRate |
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This was one of the early formulas I tried. In isolation, it is likely a better choice than the total troops produced formula. This is because it tries to factor in the temporary loss of troop count, which leaves us vulnerable to a patient enemy, as discussed above.
However it is possible to calculate how many troops the enemy (and the player) can send to a particular planet. So (I think) it is possible to calculate how much troop deficit is permitted, without resorting to a heuristic. I am not sure of this, we'll have to wait and see some games.
Quote:| If Sconibulus' method of calculating planet safety is not too complicated to incorporate, that would be excellent in that it allows slightly more aggression. Insuring home safety might still be slightly more conservative than optimal, but barely so. |
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It is possible, but I'll wait until later.
Quote:| Perhaps a better fitness function for purely safe planets would be 5*GrowthRate-3*PaybackTime, although that would almost certainly need to be empirically optimized. I think looking solely at payback time would lead to a shortsighted program that potentially grabs a lot of cheap small planets passing over those that require a more substantial investment, but pay off more ships over time. |
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Its easy to change fitness functions, so any number of them can be tried out.
Some kind of selection/optimization routines will be needed. There must be a bunch of research already done on things like this. Currently, I was thinking to have a vector of the number of ships needed from each planet, required to achieve a goal (ie troop vector). Each goal would have a value from the fitness function. So in the above example, there would be 5 goals, capturing planets A,B,C,D,E. There would be a total available troop vector, containing the number of troops available from each planet. The optimizer would have to select goals to maximize the fitness function subject to the constraint that the sum of the troop vectors does not exceed the troops available vector at any point. Is there some algorithm that will look after this?
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Fritzlein
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Re: Google AI Contest Galactic Conquest
« Reply #36 on: Sep 15th, 2010, 4:52pm » |
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on Sep 15th, 2010, 3:34pm, Sconibulus wrote:I don't think Payback time is quite as important as growth rate
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I think looking solely at payback time would lead to a shortsighted program that potentially grabs a lot of cheap small planets passing over those that require a more substantial investment, but pay off more ships over time. |
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Wait, why are many small, cheap planets worse than a few big, expensive planets? If I take five planets of growth rate 1, each with payback time 9, am I not better off than if I take one planet with growth rate 5 and a payback time of 10?
But I think you are right and my formula is wrong for a slight variation of the reason you give. I was equating two costs: the cost of having a ship in the air, and the cost of not having a ship at all. Consider the following:
Planet,Troops,Growth,Distance,PaybackTime
A,9,1,1,10
B,1,1,9,10
Both planets will repay the initial investment in ten turns, but I would much rather attack Planet B because having 2 ships in the air for 9 turns plus one turn for combat is less cost than having 10 ships unavailable, then 10, then 9, then 8, etc.
If opportunity cost is ships*turns, then attacking planet B costs me 20 ship-turns while attacking planet A costs me 65 ship-turns. In general for a neutral with growth rate G, distance D, and ships S, the opportunity cost will be approximately
(D+1)*(S+1) + (S)(S-1)/2G
At the end of that phase we will have has many ships as at the start, but our growth rate will have increased by G. So our lost ship-turns per growth (STPG) is
((D+1)*(S+1) + (S)(S-1)/2G)/G
Applying that to map 88, we get desirability ratings of
planet troops growth distance stpg
a 8 5 6 15
c 4 4 14 20
d 31 5 4 69
e 30 5 7 84
f 29 4 9 126
b 67 5 8 299
The promotion of Planet C as a desirable target is actually intuitively correct to me now that I have worked out some of the combat scenarios with these two close home worlds. The missing four ships won't tip the balance of many fights, especially since most of the planets that would be fighting are growing that much every turn. Committing 30+ ships against D or E, on the other hand, can indeed tip the balance of a fight, so the opportunity cost is real.
This raises the possibility in my mind that a correct turn 1 might be attacking planets A and C only, and keeping all other ships at home to see what the opponent does. Perhaps we will attack D or E or both on turn two, putting us slightly behind in colonization, but very likely we will be glad to have kept the flexibility to take advantage of an opponent who makes the mistake of playing too aggressively. If he takes ADE, he will lose his homeworld, whereas if he takes just AD we are not really behind him at all.
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| « Last Edit: Sep 15th, 2010, 5:16pm by Fritzlein » |
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Hippo
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Re: Google AI Contest Galactic Conquest
« Reply #37 on: Sep 15th, 2010, 5:41pm » |
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on Sep 15th, 2010, 4:52pm, Fritzlein wrote:
This raises the possibility in my mind that a correct turn 1 might be attacking planets A and C only, and keeping all other ships at home to see what the opponent does. Perhaps we will attack D or E or both on turn two, putting us slightly behind in colonization, but very likely we will be glad to have kept the flexibility to take advantage of an opponent who makes the mistake of playing too aggressively. If he takes ADE, he will lose his homeworld, whereas if he takes just AD we are not really behind him at all. |
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So we are slightly behind against AD, more behind against ACD an winning both against no charge and ADE charge. If it takes just DE ... this kind of simulations seems have to be done on the first move.
... sort the safe planets, try several subsets and tactical variants among them. Some of them would be clearly dominated, remaining should be chosen with nonzero probability.
I am not sure following turns the situation is easier.
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Oystein
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Re: Google AI Contest Galactic Conquest
« Reply #38 on: Sep 15th, 2010, 6:02pm » |
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on Sep 15th, 2010, 4:52pm, Fritzlein wrote:| This raises the possibility in my mind that a correct turn 1 might be attacking planets A and C only, and keeping all other ships at home to see what the opponent does. Perhaps we will attack D or E or both on turn two, putting us slightly behind in colonization, but very likely we will be glad to have kept the flexibility to take advantage of an opponent who makes the mistake of playing too aggressively. If he takes ADE, he will lose his homeworld, whereas if he takes just AD we are not really behind him at all. |
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Currently I believe attacking A, C and D is the best opening move.
It should beat only attacking A and C. The distance between the player seems to be great enough to compensate the lack of 32 ships. It is of course possible for the opponent to attack D the next turn, but then he will be 5 ships behind without any compensation.
The opponent may gamble and launch an unstoppable attack against my home planet, but that leaves his own home vulnerable. I can counter the next turn and one turn after he takes my home, I take his and become the only one on his side of the map.
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Fritzlein
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Re: Google AI Contest Galactic Conquest
« Reply #39 on: Sep 15th, 2010, 6:05pm » |
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on Sep 15th, 2010, 4:08pm, jdb wrote:| OMG, all this analysis for just the opening move! I'll never get the program finished. |
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As long at the complicated analysis produces simple heuristics, we're in good shape.
Quote:| However it is possible to calculate how many troops the enemy (and the player) can send to a particular planet. So (I think) it is possible to calculate how much troop deficit is permitted, without resorting to a heuristic. |
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I wholeheartedly agree with this approach. A first pass at calculating a "safe" troop level for a planet should be very straightforward, considering only ships in the air, current troop levels, and growth rates. Instead of thinking of ships moving, it is easier to think of a circle expanding outward from the planet in question, iterating by turns. On each iteration, for every friendly planet in the circle (including the planet itself), add the growth rate to your strength, while for every enemy planet in the circle, subtract its growth rate. Then expand the radius of your circle by one. Things that came into your radius are now taken into account. Subtract from your strength enemy ships already in the air toward you. Add to your strength friendly reinforcements already in the air toward you. If a friendly planet came within range, add both its troops to your strength and its growth rate to the friendly growth rate. If an enemy planet came within range, subtract its troops from your strength and subtract its growth rate from the enemy growth rate. Expand this circle a reasonable distance, say radius fifteen, and take the minimum of your strength at each of the fifteen radii as your free troops.
This doesn't get you Sconibulus's efficiency of being able to re-direct troops that are in flight, will land, and can be redirected. It doesn't account for changes in growth rate. It is therefore inaccurate, and important to cut it off from looking too far ahead, as it will get more inaccurate the further forward it looks. But the algorithm is easy and cheap and can get you a good bot up and running.
Quote:| Some kind of selection/optimization routines will be needed. There must be a bunch of research already done on things like this. Currently, I was thinking to have a vector of the number of ships needed from each planet, required to achieve a goal (ie troop vector). Each goal would have a value from the fitness function. So in the above example, there would be 5 goals, capturing planets A,B,C,D,E. There would be a total available troop vector, containing the number of troops available from each planet. The optimizer would have to select goals to maximize the fitness function subject to the constraint that the sum of the troop vectors does not exceed the troops available vector at any point. Is there some algorithm that will look after this? |
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Yeah, it's complicated. I'm trying to think of something simple that is also somewhat smart. A first pass just to get a bot up and running might prioritize goals planet-by-planet rather than globally, and in strict hierarchy rather than weighted. For example, iterate through planets in order of furthest from enemy contact to closest to enemy contact, and for each
1) Keep myself safe from ships already launched, taking into account also reinforcements launched toward me by planets further from the front.
2) Send ships to reinforce a friendly planet threatened by enemy ships already launched, but only if they will get there in time and the planet will otherwise be lost.
3) Keep myself safe from ships that haven't yet launched, in the concentric circle definition above.
4) Take an enemy planet that I can conquer by myself.
5) Take the neutral planet with the lowest stpg cost that I can conquer by myself and keep long enough for payback. (possibly more than one if ships are free)
6) Send all available ships towards a planet that is in closer contact to the enemy. (possibly the planet we should reinforce is a presently-neutral planet we are going to conquer with ships already in the air that will be close to a presently-neutral planet the opponent is going to conquer with ships already in the air)
Note that this precludes joint attacks against neutrals or enemy planets. The only cooperation is defensive reinforcements or generally centralizing ships. I think it is good enough for a first pass, though.
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| « Last Edit: Sep 15th, 2010, 6:44pm by Fritzlein » |
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Oystein
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Re: Google AI Contest Galactic Conquest
« Reply #40 on: Sep 15th, 2010, 6:12pm » |
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Or maybe we have rock-paper-scissors:
AC < ACD < ACDE < AC
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Fritzlein
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Re: Google AI Contest Galactic Conquest
« Reply #41 on: Sep 15th, 2010, 6:40pm » |
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on Sep 15th, 2010, 6:12pm, Oystein wrote:Or maybe we have rock-paper-scissors:
AC < ACD < ACDE < AC |
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Hmmm, that looks correct, except I would say
AC < ACD < ACDE <<< AC
That is to say, it would be like scissors losing to rock by one, rock losing to paper by one, but paper losing to scissors by ten. Game theoretically, you would end up almost always doing rock, or in this case, almost always attacking ACD on the first turn.
However, it seems ACD < H, i.e. taking three planets leaves the home world vulnerable. Adding that fourth strategy will force us into playing mostly AC.
I have no idea what other strategies are out there and how they affect the payout matrix. Does Hippo's ADH have a role to play, or is it dominated?
I suspect that settling on AC 100% of the time is pretty darn good, while keeping us free from coding a bimatrix solver.
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| « Last Edit: Sep 15th, 2010, 6:51pm by Fritzlein » |
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jdb
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Re: Google AI Contest Galactic Conquest
« Reply #42 on: Sep 15th, 2010, 7:33pm » |
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Quote:Yeah, it's complicated. I'm trying to think of something simple that is also somewhat smart. A first pass just to get a bot up and running might prioritize goals planet-by-planet rather than globally, and in strict hierarchy rather than weighted. For example, iterate through planets in order of furthest from enemy contact to closest to enemy contact, and for each
1) Keep myself safe from ships already launched, taking into account also reinforcements launched toward me by planets further from the front.
2) Send ships to reinforce a friendly planet threatened by enemy ships already launched, but only if they will get there in time and the planet will otherwise be lost.
3) Keep myself safe from ships that haven't yet launched, in the concentric circle definition above.
4) Take an enemy planet that I can conquer by myself.
5) Take the neutral planet with the lowest stpg cost that I can conquer by myself and keep long enough for payback. (possibly more than one if ships are free)
6) Send all available ships towards a planet that is in closer contact to the enemy. (possibly the planet we should reinforce is a presently-neutral planet we are going to conquer with ships already in the air that will be close to a presently-neutral planet the opponent is going to conquer with ships already in the air)
Note that this precludes joint attacks against neutrals or enemy planets. The only cooperation is defensive reinforcements or generally centralizing ships. I think it is good enough for a first pass, though. |
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This seems like a reasonable approach. I'll try and get this done sometime soon. Its easier to iterate through destination planets. Then its easier to launch joint attacks on a planet. So for step 1 it will look at all planets and move troops around to handle that. If there are any troops left over, it will move on to step 2 and look at all planets again. And so on until step 6.
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jdb
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Re: Google AI Contest Galactic Conquest
« Reply #43 on: Sep 15th, 2010, 7:44pm » |
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Quote:| However, it seems ACD < H, i.e. taking three planets leaves the home world vulnerable. Adding that fourth strategy will force us into playing mostly AC. |
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Quote:Currently I believe attacking A, C and D is the best opening move.
It should beat only attacking A and C. The distance between the player seems to be great enough to compensate the lack of 32 ships. It is of course possible for the opponent to attack D the next turn, but then he will be 5 ships behind without any compensation.
The opponent may gamble and launch an unstoppable attack against my home planet, but that leaves his own home vulnerable. I can counter the next turn and one turn after he takes my home, I take his and become the only one on his side of the map. |
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I don't know what will happen here. The fight could get very complicated.
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Fritzlein
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Re: Google AI Contest Galactic Conquest
« Reply #44 on: Sep 15th, 2010, 11:32pm » |
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on Sep 15th, 2010, 6:02pm, Oystein wrote:Currently I believe attacking A, C and D is the best opening move.
[...]The opponent may gamble and launch an unstoppable attack against my home planet, but that leaves his own home vulnerable. I can counter the next turn and one turn after he takes my home, I take his and become the only one on his side of the map.
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I didn't consider the counter-counter attack. I think you are right that you can take his home world, but it is slightly complicated because then he might have enough to take D from you, and he will evacuate his doomed homeworld to take A and C when your counter-counter-strike launches. He would have 65 ships left over from his attack on your homeworld after defeating your growth. You would have 8 turns of growing on D before he got there, but that's only 41 ships.
The way I count, before the growth rate evens out, you will have 5 turns of outgrowing him by 5 and 6 turns of outgrowing him by 10, for +85 ships to compensate the -39 you lost fighting neutrals. So you win that opening. Or am I missing something there? This close-range fighting is tricky.
I have come around to your idea that sending fleets against ACD is probably the best opening move. Nevertheless, proving it to be safe is awfully tricky, and I wouldn't fuss if my bot just took AC on the first move instead. Only ACD beats it, with many possible opposing mistakes, and even if he outguesses us, taking DE the second move while the opponent takes E leaves us only 5 ships behind in the colonization race. So AC is a pretty small mistake.
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| « Last Edit: Sep 15th, 2010, 11:37pm by Fritzlein » |
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