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So if the balls travel only 3x faster than the players, would be easier to dodge them?

How do players fire and hit with a guaranteed 90% accuracy? How do players fire a defensive salvo and have it hit with 90% accuracy?

It seems like the path of the balls must be tracked along some predetermined trajectory for them to take 3 or more turns to hit their target. If that's the case they are only as accurate as their aiming.

Interesting simple concept, but I don't understand the above points.

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When I first read this post I thought there is no real discussion possible.

First off what reason do they have to move? Is there a flag? At one space does defensive fire have a reaction time that prevents it being effective?

As it stands if I have a 90% chance of a hit regardless of range why move? Presumably the defensive fire is specific for incoming to the target - or possibly not. If it is specific to target then mass fire should guarantee a kill as defensive fire will be overloaded/run out.

The concept of mass fire does suggest a single controlling mind, if each player makes their own mind up anything is possible. Do they have team orders to act as one mind. Do they phase their shots to save ammo on already dead targets.

Gut feeling is that the correct tactic is for each player to fire immediately at its opposite target and hope for a kill. Also to defend against incoming whilst awaiting the result of the attacking shot. Rinse and repeat. The side that gets lucky on the first round should have an advantage that all other things being equal means it should be able to shoot defenceless targets at game end.

Obviously I have made some assumptions but my caveats could create another game from the one I am imagining. As for adding more players and more shots I am not sure it adds greatly to what seems a very boring game so will waste no more time on it. I suspect it means on millions of games the results will likely be more even as you increase numbers. Reducing to one per side will give longer running streaks ....

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It strikes me immediately that defensive fire is not worth it - 1/10 times it will miss, you lose the ball and still get hit. So you've lost the ball and the man and inflicted no casualties. This versus offensive fire - 1/10 of hte time you lose the ball, but not that man.

So if you used all your shots defensively you would fail to intercept 1/10th of 32 balls, and that 1/10th would hit 90% of the time - you would lose (on averages) about 2.9 people, and they would lose 0.

I'm sure there's higher math in there somewhere, but that's my initial impression.

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OTOH, one layer could choose to start defensively, and only fire at balls known to be going to hit.

Thus, assuming a 1:1 matchup, 'most games' would end in a draw.

If a player fires defensively, the hit chance per ball fired by the opposition drops to 0.09 (miss chance, incl interception, is 0.91).

Over 4 balls, that averages out to 0.36 hits and 3.64 misses (of which 3.24 were interceptions, and 0.4 just flat out misses). Across the entire team of 8 players that comes to 2.88 hits, and 29.12 misses (of which 3.2 are flat out misses, and the remainder interceptions). That would leave the defensive team with 3.2 balls to throw and the offensive team with no balls left to fire defensively.

3.2 balls would provide 2.88 hits ... a draw.

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Ok, to summarise:

  • Game grid ~100x100 squares
  • 8 playing pieces per team (I'll call them tanks)
  • Tanks move at 1 square per turn.
  • 4 Balls per tank (I'll call them missiles) with a max range of 12 squares.

  • Missiles move inexorably to their target (which can be tanks or other missiles in flight) at a rate of 3 squares per turn.
  • On hitting missiles have a 90% chance of destroying their target.
  • The aim is presumably to destroy the other team, and the team with the most kills wins?

I would say almost all games would end in a draw. The high odds of being destroyed by a missile would lead you to always defend yourself, and the high odds of defensive missiles being effective means that it is very likely that both sides would use up their ammo and everyone would still be alive.

The only way to avoid this outcome would be to have a localised two-on-one matchup where a tank burns up its ammo against its two opponents who still end up with ammo to make the kill. But then the survivors of this battle would have to face the other players, and every tank who uses an extra missile to kill someone will end up with one less to defend themselves, so making a kill would be a death sentence. Out of 16 attacking shots vs 16 defending shots it is likely that only one or two will result in a death due to the roll of the dice. Whoever beats the odds and gets a kill will win. If one or both teams were stationary, the outcome would be the same. In fact just thinking about it, whoever gets first kill can destroy some enemy ammo on board the tank, so one kill will result in a snowball effect too, leaving the other team less ammo to defend themselves with.

Reducing the maximum range makes things a little more interesting, if you can get two tanks in range and make a missile burn out without hitting you will have an ammo advantage. This is where the tactics might come in, setting up localised firefights hoping that you can entice your opponent to fire at a range where you can outrun their missiles. Winning the game might depend on your opponent making a silly mistake.

Games like chess and command and conquer set up different styles of units for just this reason. Identical matchups can result in everyone on the board dying or in this case maybe nobody on the board dying.

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It isn't as simple as the offense dominance would make it appear. Yes in principle a ball shot offensively is superior to one shot defensively, but each player has 4 and you don't need to hit each enemy player 3.6 times. You are in overkill territory firing all of them offensively, if both sides do.

If one side fires all shots offensively in the first couple turns, the other side can expect to win by firing 2 at each enemy and using all the remaining shots to target only the incoming headed for a quarter of the own-side team, one shot each. At 2 each, there is only a 1% chance any defender on the all-offensive side survives, and the expectation is no survivors, with only a low chance of a single survivor. The half defense team, on the other hand, will lose its undefended 6 members with .9999 probability, but that is deep into overkill. It two protected members face only a 32% chance of being hit each, with an expected survivors of 1.5 team members.

You can prevent any loss of shots due to hits by simply firing off all shots before any enemy ones can arrive. However, firing early to avoid losses to hits also means losing the ability to adapt to the outcome of previous shots (by e.g. reducing overkill, or by allocating unneeded offensive shots to defense).

Movement is relatively trivial, but can potentially give some benefit to a slower firing side that backs up those it is defending. For example, suppose the half-defense team above also fires half its shots slower. It can trade a second shot at an enemy (who can have no defense), which raises surviving enemy by only 0.09 in expectation, if one of the defending shots is seen to miss its assigned incoming, since there an extra second defense shot at the incoming raises own-side survivors by about 0.60 in expectation.

Point being, there is exploitable value in seeing both the enemy's prior moves and the outcome of prior shots, in allocating your own. This creates some incentive to fire slower. Though not enough to risk an unintercepted hit that would eliminate several shots before they can be taken.

To game theory it out, you would need to first find the game's Nash equilibrium in expected returns for shot assignment, terms. Starting with a simplification that abstracts from movement and regards shot-assignments as simultaneous, only assigning to offensive or defense, first find the Pareto optimal assignment of offensive and defensive shots. Since the game is symmetric this will be a symmetric allocation - there is some portion of shots the "best player" "should" assign to defense, with those simplifying assumptions.

It will get more complicated once you impose actual sequential firing, though. Firing first appears to be dominated (lol) - you are giving up the ability to exploit potential reaction to prior enemy decisions, and not getting anything in expectation, in return - assuming the initial range is long enough that you can't get a positive chance of eliminating enemy shots before he can fire them off.

By the rules that is satisfied as long as you are 3 or more apart (there isn't any rule mentioned about only being able to fire off one "ball" per turn - if there were, the minimum safe distance before there is some incentive to fire first would be 9).

Op problems aren't as trivial as they can sometimes first appear...

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Let us start with the following reduced form of the game.

Each side has 8 targets and 32 shots. Each shot can be assigned to attack one of the enemy targets or to defend one of the friendly targets. All assignments are made simultaneously and secretly at the start and the execution then follows as one resolution phase. There is no movement. The allocation decision is to assign your 32 shots to each of their 16 possible uses.

Since that already has really lots of possibilities, let's first simplify even further to just 3 shooters per side. There are 6188 possible allocations - per side - and 38.3 million possible "games". (6188 is the number of compositions of 12 into 6 bins, with 0s allowed). Actually the number of functionally distinct allocations is lower - that figure treats positions as mattering, as though the players all have names and we care which one lives (lol).

We will assume that the defensive fire is minimally "efficient", in the sense that each targeting incoming is "tried" with a defending shot and the outcome is known before the next one defending that same target is used. In other words, there is no defending "overkill" of the same destroyed incoming, if other incomings still remain targeted at that same friendly. But there can be defending overkill by assigning defensive shots to a friendly who isn't targeted (though that wouldn't happen in the original, sequential game, where defensive allocation decisions come after at least one shot at that target etc).

After writing a little simulator of that game in Mathematica, with all possible strategies numbered, I ran a search for the strategies that on average over 20 games beat "shoot 4 at each target, no defense". 1094 out of 6188 strategies beat that attack.

Next I ran the search using a candidate optimum strategy of 2 attacks assigned to each target, and 6 defense assigned to 1 of your own (which was one of those superior to "even all attack"). There were 2166 that beat that strategy. Next I ran against attack {2,2,2} with defense {3,3,0}. 1320 beat that strategy. The intersection that beat all 3 of the above was still 179 strategies long.

I added the defense heavy {1,1,1} attack with {3,3,3} defense, which 2461 strategies beat, and that left an intersection that defeated all of the above that was 45 strategies long. It included assymetric attacks and defenses, and the neutral {2,2,2} attack, {2,2,2} defense. 899 strategies beat that choice, but to beat all of the above so far winnows us down to 8 strategies. 5 of these are permutations of {3,2,1}, {3,2,1} in various random orders, 2 of them are 8-4 splits, one each offensive and defensive, and a single case of a 5 attack 7 defense.

Of course all those can be beaten in turn by large subsets of the strategy space. The 3-2-1 splits did well against my "battery" of even splits because they provide adequate defense against their various levels of attack, saving more against weak attacks and less against strong ones, while having a decent chance of putting their own strongest attack up against a defense they could overpower, while also having a chance to pick up easy hits on undefended enemies, versus the concentrated-defense or all-attack opponents.

In game theoretic terms, this is already enough to indicate that the best strategy will be a mixed one, meaning something less than fully predictable and incorporating some randomness. Rather than a single fixed optimum, it is a paper scissors rock situation only more so - with our simultaneous choice simplification, to be sure.

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The usefulness of defensive shots may actually increase compared to that simplification, in the real game. The rules on speeds mean in effect that there will be information about the outcome of defensive shots much faster than information arrives about the outcome of offensive shots. Defensive shots will not really be wasted trying to defend targets not subject to lots of incoming.

Basically, the defender can choose to concentrate his defensive fire to just barely save whoever the hit-chances he actually achieves, allows him to save. This effectively eliminates defensive overkill, while offensive overkill remains a problem.

Worse, defensive shots will be allocated on the most weakly targeted friendlies, if the offensive shots come in unevenly distributed. Any "lumpiness" in the attack profile, therefore, is an invitation to allow the most heavily targeted to be "overkilled", while saving the less heavily targeted. So the defender can actively boost offensive fire overkill inefficiency.

Next comes the issue of whether spatial maneuvering can help at all, compared to staying completely clumped. I think the effect is minimal if the target of each shot is known from the minute it is launched. If on the other hand only the direction the shot "chooses" to move gives any indication of its targeting, then some spatial separation can be useful for a side expecting to concentrate defensive shots to just barely save as many of its own side as possible. You'll know which incomings to shoot at.

You might think you'd have to worry about not having enough defensive shots in range if you spread, but it won't be a problem. Anyone targeted by more than the 4 shots he can defend against himself is probably going to be in the overkilled category. At most you might want to use pairs. Notice all that only applies if the rules don't immediately tell you who each shot is targeted at, the minute it is launched - if they do, then spatial spread isn't needed.

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Dueling theory holds, on average, that the force, assuming equal capability and size, which fires first will ultimately win the duel. This is precisely what the Army used to teach in FM 100-5 Operations in summarizing the key principles of the AirLand Battle and the modern battlefield. Within the context of this overall scheme, it appears we're now talking about assessing the effects of varying the firing doctrine. I did this sort of thing a bit many moons ago when looking at FAD (Fleet Air Defense) vs. incoming hordes of Russian supersonic cruise missiles. Depending on refire rates and warship SAM loadouts, you could examine the ups and downs of, say, SLS (Shoot, Look, Shoot), useful when there's sufficient battle space (threats relatively far away), as compared to the scarier SSL (Shoot, Shoot, Look), the natural result of battle space compression. In the latter case, the object's to get as many SAMs out as possible to the targets, then see what happened. Remember, the missile's are coming in at Mach 3.5, and the SAMs have flyout times, launcher reload times, and maybe even dedicated shipboard guidance radar issues to contend with.

This game assumes what we used to call perfect (target) assignment, meaning each firer engages his own discrete target with no overlap from other defenders; further, each offensive shot has PH (Probability of Hit--really it's not a probability but a certainty) of 1 and an SSPK (Single Shot Probability of Kill) of 0.9, with unknowns consisting of time from target acquisition until firing, weapon time of flight, reload time, offensive decision criteria as to when to go defensive (by individual shooter or on order from group leader), target selection logic if initial attack removes one or more opponents from play, firing doctrine initially and as the engagement unfolds, etc. In many ways, the actual FAD problem is far more tractable, there being far fewer unknowns and very real technology limits, especially before VLS (Vertical Launcher System) tech arrived, on who could do what and how fast, seeing as how only a limited number of launchers per ship per available, and reloading was an important time issue. In the game, there are few fewer bounds on the problem and lots less data to work from. Now, armed with a charge number, a gym, some dodgeball teams and a little time, I could rapidly produce some data with which to bracket the problem!

Regards,

John Kettler

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I don't really understand all the maths, but don't forget that once a playing piece/tank is killed, all their missiles are destroyed too.

Each offensive shot that is not defended will compound the odds in the favour of whoever is lucky enough to make the first kill.

So choosing not to defend seems like a terrible strategy, whoever fires first will win, as their missiles will hit first, and leave them time to defend against their opponent's first salvo. There will be no second salvo, because the high kill chance means the whole opposing team is dead.

I think Jason says it above, with equal limited ammo for each team, there is no advantage to overkill on one target, or manouvering. As long as you play the odds and defend against every offensive shot, the game will come down to a roll of the dice.

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No, the first shooter doesn't kill all the enemies before they can get their shots off. The shots are quite slow, actually, and the targeted friendlies can easily fire off their entire ammo load before anything fired at them actually arrives and resolves its to-hit chance.

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Yes, 3 "boats" per side, with 4 "missiles" per boat, 12 missiles to allocate total.

Each missile can be put to one of 6 uses - shoot at enemy 1, 2, or 3, or protect friendly 1, 2 or 3. Both sides pick their strategy simultaneously and secretly at set up - meaning allocate all of their shots, both offensive and defensive.

I encode that as two lists of numbers 3 long, offensive shots first. So {{2,2,2},{3,3,0} means shoot two missiles at each enemy, while protecting only your first two "boats" with the remaining missiles, 3 each.

The possible strategies of this type are the ways of dividing the 12 missiles among the 6 uses, with 0s allowed. Which is a combinatoric function called Compositions[12,6] (or IntegerDecompositions in a later version of Mathematica). So a strategy is fully specified as one of those decompositions, and a "game" as two such played against each other.

What I mean by the paper scissors rock aspect is the best number of shots to allocate to offensive shots depends on the strength the enemy spends on defensive shots.

For example, if the enemy uses no defensive shots, the best offensive allocation is just 1 shot to each enemy. Because those get a 90% chance of a kill, while a second at the same target only add 9%, when there is no defense. Meanwhile an enemy who has allocated nothing to defensive is necessarily launching big overloaded attacks, making extra defensive shots quite useful - typically 81% chance of saving a friendly. So a good counter to a {4,4,4},{0,0,0} all-out attack strategy, is something like {1,1,1},{5,4,0} - minimal but efficient attack, combined with strong defense of some of your own, without trying to cover them all.

But that same {1,1,1},{5,4,0} would likely lose to {7,1,1},{1,1,1}. The weak attack can be effectively countered with a weak defense - the first gets only a 9% shot at 3 targets, likely getting none (or 1 sometimes) while the response gets an overloaded attack on the first enemy and an open shot at the last, likely killing 2.

The point is, the right mix of offense vs. defense and of concentrated or spread out attack or defense, depends in "head fake" fashion on the opponent's decisions about the same questions. Just as you can find a paper to beat rock and a rock to beat scissors, with cyclic superiorities but no one best.

Nash game theory tells us that the optimal strategy in such cases is a mixed strategy rather than a pure one. Meaning, play strategy A with probability p and B with probability q etc, through some whole set of possibles, deliberately randomizing the choice to remain unpredictable. (A strategy is mixed if it randomly picks among possibles, and pure if it always plays the same move - or in a sequential game, if it always plays the same move given the prior moves by both sides).

The same principles will apply to the larger games, there are just already enough possible allocations with 3 that running them all against each other can take minutes. The full original game size you gave, you could play an instance of it fast enough, but there are millions of possible shot-allocations in that one, and the number of possible games is the number of strategies squared (yours, and mine, varying independently) - so you couldn't exhaust the bigger version with reasonable computational effort.

As for lowering the hit chances, it will tend to make offensive shots more effective, because it reduces the potential overkill to allocate to defense, and also because the probability edge of attacking vs. defending is going as the hit chance (hit chance straight for offense, hit chance squared for a defending shot - since those only help if the attacking would have hit, and the defending hits the incoming). Drop the hit chance low enough and the optimum will be to fire all the shots offensively, and to spread them evenly.

All, in the simplified version of the game given above, that is. In the full sequential version there is a whole additional issue of trying to maximize the benefit of information about enemy allocations and the results of prior shots, to choose where and how to fire your remaining shots, better.

I hope this is interesting.

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