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Game Theory
"Never interrupt your enemy when he is making a mistake." This is a very simplified version of what game theory is. People get PhD's in game theory and work for governments, deciding how to fight real wars. However, if professional game theorists work on real wars, then amateur game theorists can utilize it to send their toy soldiers into hobby battles. In its most basic sense, Game Theory is about deciding upon courses of action, based upon expected results. In this way, you cannot make the correct decision about a course of action unless you have some idea of what is going to happen as a result of that action. There are different sorts of games. Games like chess have all the pieces in front of you, and you know the capabilities of each piece. This is said to be perfect information. Some games, like poker, have hidden information (your opponent's cards) and so you have to make your decisions based on the information you do have, and the probabilities of what the hidden information may be. Warhammer is slightly different. You have all the information about what the pieces are capable of, but the dice have a randomizing effect, so you have to account for probabilities when making your choices. The more information that you have, or the less the results are left to chance, the further into the game you can predict, and then use those predicitions to base your choice of action on. For instance, in Chess, a game of perfect information, Grandmasters - and computers - can often see 5 or 6 moves ahead. They can predict how you will move before you move, and therefore set up traps for you. In games like poker, you cannot see ahead at all. In Texas Hold'Em, the entire game can change based on the next card dealt. As such, betting is done based on the cards you currently have, and the odds that the next card will help your hand compared to the size of the pot, with no planning for three cards away. If you can bet your opponent out of the pot now, you don't have to worry about what comes later :) In Warhammer, you can make decisions about future moves with a direct proportion to the odds that your choice of action will succeed. Meaning, if I shoot a unit, expecting to kill it, and use that expectation to predict that my opponent will then retreat, allowing my units to advance the following turn, if my shooting does in fact destroy the enemy unit, I'm going to be pretty correct about my opponent's response. If the unit I shot at is not destroyed, his response is going to be different however. So, if I have a 70% chance of destroying the unit, I can guess that his other units will fall back 70% of the time. Yesterday, we examined the math required to compute probabilities, so today, we will examine how to use those probabilities when making decisions. Only one correct decision? Well, that's hard to say, because there are different approaches to game theory. There are generally going to be two correct decisions, which correspond to the two approaches to game theory. Classical Game TheoryThe first approach is refered to as classical game theory, because it was the first approach studied, and because it's what you're going to use most of the time. Plus, that's just what it's called. Classical game theory dictates that you study the expected outcomes from any decision you make, and take the one with the best expected upside, or least expected downside. A simple example: I have a unit of 12 Firewarriors. (S 3, T 3, BS 3, WS 2, I 2, A 1, W 1, Sv 4+, Str 5 Rapid Fire Gun 10 points) My opponent has a unit of space marines (S 4, T 4, BS 4, WS 4, I 4, A 1, W 1, Sv 3+, Str 4 Rapid Fire Gun, 15 points). I find my firewarriors six inches away from the marines. Should I charge the marines, or should I shoot at them? (Rapid fire weapons don't allow you to charge if you fired). If I shoot, I have 24 shots, that hit 1/2 the time, and cause wounds 2/3 of the time. The marines will fail 1/3rd of the wounds. So, 24 * 1/2 * 2/3 * 1/3 = 8/3 dead marines. At 15 points per marine, that's 40 points of damage inflicted. If I charge, the marines have higher initiative, and strike first. They get 10 attacks, hitting 2/3rds of the time, wounding 2/3rds of the time, and I fail 1/2 my saves. So, before I strike, the marines kill off 2.22 firewarriors. The remaining 9.77 firewarriors attack back with 2 attacks each. They hit 1/2 the time, cause wounds 2/6 of the time, and the marines fail their saves 1/3 of the time, so 9.77 * 2 * 1/2 * 2/6 * 1/3 = 1.08 dead marines. I've inflicted 16 points of damage on the marines, while they have inflicted 22 points of damage on me, for a net of -6 points. Clearly, a choice between dealing 40 points of damage and taking -6 points of casualties indicates that my firewarriors should shoot the marines. This sort of math can be applied to all sorts of situations that occur during designing your army list, and playing the game. If you know what your opponent will be playing, you can select the weapon options that give you the highest expected returns, if you do a little math. You can select the units that will be the most effective against their army. A more complex and relevant example that might actually occur during a game is in choosing which unit should shoot at which enemy. Imagine that you have three squads of marines. One has 10 marines with bolters, one has 6 marines with bolters, and one has 6 marines, 5 with bolters, and one with a heavy bolter. Up run these 10 great ugly green orks, intent on getting into combat with your marines the next turn. This turn, they end up 9 inches away, just short of their charge range. You don't want to fight the orks in close combat, so you have to destroy them all this turn. Further back, there are some other orks that you would like to kill off too, but they're not the immediate threat. Which unit should you fire with first? See, this is the sort of problem that will actually occur. You want to kill all the orks, but you don't want to overkill them, because then you're wasting your firepower. So, what you want to do is try and figure out how effective each squad would be. And, to do this in a way that would be applicable to doing during a game, we're going to use a lot of approximation this time. We're going to round down fractions to give us wiggle room in planning. Squad A (10 marines): 10 marines get 20 shots. They hit 2/3rds of the time. So, approximately 2/3rds of 20 is 12. 12 hits, wound the orks half the time, so that's six wounds. And, orks don't get saves against bolters, so squad A can be expected to kill six orks. Squad B (6 marines): 6 marines get 12 shots. 2/3rds of 12 is 8 hits. Half of 8 is 4 wounds, and again the orks don't get saves against bolters, so squad B can be expected to kill 4 orks. Squad C (5 marines + marine w/ Heavy bolter). 5 marines get 10 shots. 2/3rds of 10 is 6 hits, and half of that is 3 wounds. The Heavy Bolter gets 3 shots, 2/3rds of which hit, yielding 2 hits, and 2/3rds of those kill an ork without a save, so the heavy bolter can also account for 1 kill. Squad C can also account for 4 kills. So, who should fire first? Either squads A and B or squads A and C would probably kill all 10 orks, leaving either squad B or C free to fire at the orks in the distance. And since Squad C has the heavy bolter with the extended range, you probably want to count on squads A and B to kill the orks that are assaulting. But we still don't know which of A or B should fire first, so we'll leave that for a while, and lead to... Romantic Game TheoryRomantic game theory has little to do with the sort of adult novelty games that you find in Spencer's Gifts. It's called romantic because it has to do with exploring what choices you should make based on having the best things happen. For instance, if your opponent has a monolith, and a squad of necrons, and you have a squad of 10 marines, with a rocket launcher, should you target the monolith, or the necrons? Classical game theory says that you will get more points if you attack the necrons, because the odds of destroying the monolith are low. But Romantic game theory says that you should target the monolith, because if you succeed, it's worth a lot more points, and can cripple your opponent's plans. Sometimes, romantic game theory is the only thing you can use, because taking a classical approach will lead you into a course of action where you cannot win the game. In our previous example, if you always take the classical approach, you will always shoot necrons and never shoot the monolith. But if you never shoot at the monolith, it will cause you problems over and over. So, while the expected results of shooting at the monolith are worse, never shooting at the monolith may cause you serious problems. The existance of the Lottery speaks to how much faith some people put in romantic game theory. Most real decisions are made, subconsciously, with a combination of the two approaches, and both have applications to the game we're playing. Back to our question about shooting at orks. What happens if we adjust the numbers a little to account for some romanticism? We don't want to go all out, and say all our shots will hit, and all our hits will wound, but, to take a romantic view, what if we assume that slightly more of our shots will hit than average (say, 7/9 instead of 6/9) and we assume that slightly more of our hits will wound than average (5/8 instead of 4/8). We'll also round UP our estimates. With these assumptions, lets get some best-case scenarios Squad A (10 marines w/ bolters) will have 20 shots, and will get 16 hits. Of those 15 hits, 10 will kill orks. Squad B (6 marines w/ bolters) will have 12 shots, and will 9 hits. That will result in 6 kills. Squad C (5 marines + heavy bolter) will have 10 bolter shots, hitting 8, killing 5, while the heavy bolter will account for 2 orks, for a total of 7 kills.
From this, we can see that squad A might come pretty close to wiping out the squad of 10 orks alone. If that romantic notion does happen to come true, then both squads B and C would be free to fire at the orks in the distance. However, as backup, even with a classical approach, we would expect that the combined firing of A and B would handle 10 orks. So, when we mix our romantic expectations into our planning, we see that squad A should fire first, because they might get the job done alone. Squad B should fire second, because they're expected to do as well as squad C (who has greater range for addressing other threats), and squad C should be held as the just-in-case unit, but we're planning on having them shoot downfield. Note: Just because we plan to have them shoot downfield, doesn't mean they should shoot downfield first. Dice have a way of betraying us, and expectations don't always work. If squad C does fire downfield first, and A and B fail to wipe out the assaulting orks, then it is too late to have them do clean-up duty. So, in summary, the classical approach deals with what you expect to happen, while the romantic approach deals with what you hope will happen. Both have value when you have to make a choice about what to do next. |