Counterintuitions and the well-ordering theorem
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From: Daryl McCullough on 27 Oct 2009 10:59 Herman Jurjus says... > >Daryl McCullough wrote: >> Bill Taylor says... >> >>>> I know that you're not offering that as a mathematical argument, >>>> but I would like to understand your intuitions here. >>>> "Either player 1 has a winning strategy, or 2 has a defense" >>>> needs a little more argument to be compelling, because of games >>>> like "Rock, paper, scissors" with no strategy guaranteed to win. >>> OUCH! Daryl, this is not up to your usual high debating standards. >>> RockPS, like almost any game of incomplete information, has no >>> "guaranteed winning strategy", as you say, but AD games are >>> not of this type. >> >> Actually, could you explain in what sense chess or checkers are >> games of complete information, and rock/paper/scissors is *not*? > >Now it's my turn to say 'OUCH!' >Can you explain in what way chess or checkers are /not/ games of >complete information? I just explained what I meant. In ALL games, the unknown is what your opponent will do. That's true for chess, checkers, rock/paper/scissors. In the case of games of incomplete information, there is ADDITIONAL information to be had (the contents of hidden cards, in a card game, for example) BESIDES your opponent's future choices. >> In all cases, each player must make his move in ignorance of >> what move his opponent will make. There is no "hidden state" >> (such as the hidden playing cards in a game of cards), unless >> you count the intentions of your opponent as hidden state, in >> which case, all games are games of incomplete information. > >Only if you mis-characterize them as 'both players simultaneously choose >a strategy, and keep following these'. But that's not what happens when >two people play chess. Mathematically, it doesn't make any difference whether you choose a strategy all at once, or choose it move by move, taking into account what your opponent played. There is, of course, a *practical* difference, in that the computational difficulty of thinking many moves ahead means that in practice, you adopt a short-term strategy, and then revise it in light of new information. But surely you agree that it makes no mathematical difference, right? The short-term strategy modified in light of developments is mathematically equivalent to a particular long-term strategy. Anyway, if you are talking about the plausibility of determinism, it certainly isn't people's experience that chess has a winning strategy. It might be mathematically provable, but it's not a fact that is self-evident, and it's not a fact that comes into play much in actual games. >It really /does/ make a big difference whether a game is of complete >information or not. You haven't defined what "a game of complete/incomplete information" means. The Wikipedia definition says: "Complete information is a term used in economics and game theory to describe an economic situation or game in which knowledge about other market participants or players is available to all participants. Every player knows the payoffs and strategies available to other players." It seems to me that by that definition, rock/paper/scissors IS a game of complete information. The difference with a game like chess is not whether its complete information or not, but whether the moves are alternating or simultaneous. >Look, there may be any amount of valid reasons to be suspicious of AD, >or to dislike it, or whatever. But your objection is not one of them, imho. Well, certainly you would agree that IF the infinite game is played in the following way, then it is the same situation as rock/paper/scissors: Each player chooses a strategy, which is a deterministic function from finite sequences of naturals to finite sequences of naturals. Then we compute an infinite sequence by interleaving applications of the two strategies. If the result is in the given set, the first player wins, otherwise, the second player wins. With this description, there is no reason to believe that there is a winning strategy for either player, any more than there is for rock/paper/scissors. You might not like to think of it in terms of choosing a strategy, rather than choosing naturals one at a time, but I can't see, mathematically, how it could make any difference. Maybe you can explain that? -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 27 Oct 2009 11:04 Herman Jurjus says... >Might i humbly suggest that that is explainable by the fact that many >people are confronted with choice in the earliest stages of their >studies, and not with AD? > >And what would have happened if ZFC+AD had not been inconsistent? Or if >the inconsistency had not been discovered in the 1950s, but, say, in 2005? I don't understand why you think that "all games are determined" is intuitively true. It's not *obvious* that chess or checkers has a winning strategy; it's *provable*. To prove it, you have to use the fact that they are finite-length games (actually, the possibility of ties or cycles makes chess not precisely a finite sequential game, but you can change the question to: is there some strategy that can guarantee avoiding a loss?) What reason is there for believing that the principle applies to games for which is not provable? -- Daryl McCullough Ithaca, NY
From: Herman Jurjus on 27 Oct 2009 12:05 Daryl McCullough wrote: > Well, certainly you would agree that IF the infinite game is played > in the following way, then it is the same situation as rock/paper/scissors: > > Each player chooses a strategy, which is a deterministic function > from finite sequences of naturals to finite sequences of naturals. > Then we compute an infinite sequence by interleaving applications > of the two strategies. If the result is in the given set, the first > player wins, otherwise, the second player wins. > > With this description, there is no reason to believe that there is > a winning strategy for either player, any more than there is for > rock/paper/scissors. You might not like to think of it in terms > of choosing a strategy, rather than choosing naturals one at a > time, but I can't see, mathematically, how it could make any > difference. Maybe you can explain that? So we have two descriptions, which you claim to be equivalent. (I'm not so sure that the two are conceptually the same, but let's leave that, for now.) With one of these descriptions, a certain conclusion is not self-evident, with the other it is (at least for me; you don't agree with the latter). For me, the equivalence between the situations would lead me to accept the conclusion in the second case as well. You, on the other hand, seem to reject the conclusion in the first case, on the grounds that it is not self-evident in the second case. I cannot look inside your brain, but could it be that, so far, you simply /didn't/ intuitively evaluate the first description for yourself at all? I.e. that you just replaced it in your mind with the second description, and judged the situation based on that second description, while further totally ignoring the first description? [Because, in retrospect, that's what /i/ always did when thinking about AD, in the past.] -- Cheers, Herman Jurjus
From: Butch Malahide on 27 Oct 2009 12:30 On Oct 27, 11:17 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > > Well, don't look at me; i said right from the start that both AD and AC > are evidently true. Not much use, then, in trying to use reductio ad absurdum on you, then, is there? Actually, my comment was directed to Bill Taylor, who is going to ignore it because he doesn't have a good answer.
From: Daryl McCullough on 27 Oct 2009 12:59
Herman Jurjus says... >I cannot look inside your brain, but could it be that, so far, you >simply /didn't/ intuitively evaluate the first description for yourself >at all? I.e. that you just replaced it in your mind with the second >description, and judged the situation based on that second description, >while further totally ignoring the first description? You're right, that I substituted an alternative description that I thought was more mathematically tractable to reason about. So I'll try to stay to the first description: First player selects a natural, then the second player selects a natural in response, then back to the first player, etc. But I don't understand how this description intuitively suggests that one player or the other has a winning strategy. In a finite game, we can always in theory work backwards from a winning position. Let W_0 be the set of winning positions for the first player. Back up two moves and let W_1 be the set of positions such that there is a move by the first player such that no matter what countermove the second player makes, he will end up in a position in W_0. Keep backing up to form W_0, W_1, W_2, etc. Now, let W = the union of all the W_i. This is the set of all positions such that it is possible for the first player to force a win, starting in that position. If the starting position is included in set W, then the first player can force a win. If not, then the second player can *avoid* the positions in W. In other words, the second player can avoid losing. But in a finite game, if the second player avoids losing for long enough, then he wins. This argument doesn't work for infinite games, because there is no way to "back up" from a winning position. -- Daryl McCullough Ithaca, NY |