Well, yes, as this applies to any deterministic game, correct?

What is the reference of "this"?

This reply:


Assuming it is inconsequential if any player has a strategy that is infallible. And, if they do, then the game is no longer worth playing if no winning strategy can be entertained as a player with (n+1) move, with the player with the first move always winning.

Sorry for being dense, but I don't understand what property you're referring to in your last post. It seems to be talking about conditions for a game not to be worth playing, but I'm afraid I just didn't get it.

Let me be simple. If a player can't win a game, then what's the point of playing it?

I mean, everyone wants to start out as white when playing chess if there's money or financial reward for the simple fact of winning, correct?

I suppose there could be other motives, such as boredom, incredulity, or simply to better understand why you can't win. Incidentally, note that the existence of a winning strategy for one of the players does not mean that in an actual, particular play, one of the players is guaranteed to win. Maybe he doesn't know the strategy, maybe nobody knows the strategy, maybe he knows the strategy but has forgotten it, maybe there is a strategy but nobody can follow it because it is too complicated, etc.

As for chess, I don't think everyone wants to start out as white. For some time I personally was more comfortable playing black, and I think many players don't mind playing either way. And in any case, it's not clear that there is a winning strategy for white. Either black or white has a strategy for not losing (i.e. at worst to force a draw), but nobody knows which. It could be black's.

Well, think of it analogously towards such things where game theory is applied, such as warfare scenarios or nuclear warfare.

Does that help?

No, it does not help at all. I'm beginning to lose track of what is your point. At first, I thought you were (erroneously) claiming that there can be no winning strategies for any games, since, supposedly, players could readily adapt to any such strategy through rote training or whatever. Now you seem to be claiming that a game with a winning strategy for one of the players would be pointless to play, to which I replied that (i) the existence of a winning strategy does not imply that a particular play is determined, since, for a variety of reasons, it could be that the player with the winning strategy has no access to it and (ii) in any case people could want to play the game for a variety of reasons other than winning. Finally, you seem to claim in your last post that there is some kind of problem with applied game theory, but you don't indicate what the problem is or how it is remotely related to the fact that some games (like Hex) have a winning strategy for one of the players. So I'm kind of lost...

Finally, you seem to claim in your last post that there is some kind of problem with applied game theory, but you don't indicate what the problem is or how it is remotely related to the fact that some games (like Hex) have a winning strategy for one of the players. So I'm kind of lost... — Nagase

Yeah, this is pretty much what I'm getting at. Namely, the nonsense of applying game theory to real world problems.

Let me elaborate with a real life example.

Let's take the case of Mutually Assured Destruction. If game theory leads us to assure a suicide pact between two opposing nations, then we ought to reject it, or not?

Reject the pact or reject the applicability of game theory?

In any case, as with any mathematical formalism, game theory provides an ideal model of certain situations which involve strategic behavior. It is a model, in that it serves to highlight certain causal or structural dependencies of a given phenomenon (in this case, strategic behavior); it is ideal, because it involves a deliberate falsification of reality for simplification purposes (cf., among many others, the work of Nancy Cartwright and of Angela Potochnik for more on this). So I don't think applying game theory to reality is anymore nonsense than applying the Lotka-Volterra equations to study population equilibrium or the use of infinite populations in certain models to screen off genetic drift considerations, or even the application of ideal gas laws to explain the behavior of gases.

Yes, well we actually have a working model of game theory in practice in perhaps its most extreme form, being Mutually Assured Destruction.

Again the point about super-rational players is an inconvenient truth about the applicability of game theory manifest, yes?

Nobody likes super-rational players for the matter, as they can't be reasoned within, or from, dominant strategies.

I really don't understand what you're getting at. What exactly is your point?

Sorry to interrupt your discussion, but even though I am a retired math prof I continue to learn about mathematics results by following some of these threads. My thanks to the participants.

Fishfry in particular has opened my eyes to modern set theory, but others have as well. And for game theory, I knew that Nash had used attractive fixed points but I now learn he employed a result I was unaware of, a set theory extension of Brouwer's Fixed Point theorem by Kakutani:

https://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem

Brouwer's Theorem provides an existence result, but doesn't give an algorithm for reaching this point. I am quite familiar with Banach's Fixed Point Theorem (having generalized his result for infinite compositions of functions rather than iteration of a single function - there are dozens of generalizations!) which does describe a simple algorithm.

( https://www.coloradomesa.edu/math-stat/catcf/papers/banach-extension-theorem.pdf }

OK, I'm done. :cool:

I really don't understand what you're getting at. What exactly is your point? — Nagase

That through enough iterations in any deterministic game, then advantageous situations are known prior to making a decision on the decision tree, and hence, the chance of winning becomes very small.

I hope that made some sense.

I've already replied to this in my second post in this thread: one of the players may be able to force the other to perform certain moves, or perhaps the other player's moves are somehow irrelevant (think of forced checkmates). That is, it may be that there are no advantageous situations in the decision tree of one of the players to be known.

Yes, so if that's the case that everyone wants to start out as white in chess, because there is a natural advantage to starting as white, then the game becomes meaningless for both players if an assured victory can always be entertained as white.

Notice that a perpetual stalemate is tantamount to the above.

But we don't know whether it is white that has the winning strategy. It may be black (or neither).

Statistically, white wins the majority of games against black. But, that's irrelevant to the point I'm trying to make about given enough iterations that both players at best would be able to enjoy a stalemate.

The case with humans, comparatively to warfare is that if both players have an absolute deterrent, then the notion of a zero sum game becomes irrelevant. Furthermore, if one of the players makes a mistake, then that spells doom for us all...

But it is not a matter of statistics, it is a matter of whether there is a winning strategy or not. For all we know about chess, maybe white has a winning strategy, maybe black, maybe neither (that is, maybe the best strategy leads to draw).

Yes, and all that hinges on whether the other player is;

1. Less informed.
2. Less rational.
3. Less motivated.

Once you have enough iterations and sufficiently motivated players, then whoever has the first move, will dominate the game if we strictly are talking about chess. I don't think the same applies for Hex, as you mentioned.

If you apply this same line of reasoning to such instances where both players have to be MORE, motivated, rational, and informed, then you have super-rational players.

Well obviously the reason why people enjoy playing Chess is because its outcomes are uncertain due to players bounded rationality and tendencies to make mistakes, assuming that the skill difference between opponents is roughly even. This is especially the case for the variant Chess960 that in being randomly initialised diminishes the role of opening-theory. The Chess community are well aware that the rules of Chess have to evolve if Chess is to remain an interesting non-predetermined spectacle. Perhaps the game will continue to fragment into more and more alternatives. Personally, I think there are more interesting board-games to professionalise.

First, the psychological conditions of the players are irrelevant. Either there is, or there isn't a winning strategy. This can be determined entirely by analyzing the space of the play, and is independent of such extrinsic factors.

Second, your claim that the first player will "dominate" the game is, again, pure speculation. We don't know that! If you do, I suggest that you write your proof and send it to a reputable journal on Game Theory.

Once you have enough iterations and sufficiently motivated players, then whoever has the first move, will dominate the game if we strictly are talking about chess. — Shawn

We don't know if that's true of chess but it is true of Connect 4. I used to play against a computer that played a perfect game to try to learn what moves I should do when playing other people (I played a lot of Connect 4 in Thailand). Sometimes people like to play even if they're guaranteed to lose.

Second, your claim that the first player will "dominate" the game is, again, pure speculation. — Nagase

https://www.chessgames.com/chessstats.html

Years covered: 1475 to 2020 (546 years)
All time controls (946,291 games)
White wins 357,549 games (37.78%)
Black wins 266,196 games (28.13%)
322,520 games are drawn (34.08%)

Again, statistics are irrelevant for a mathematical proof that white has a winning strategy! I don't know what else to say in this regard.

Why is that so? Quite an interesting subject...

I would like to reiterate my thoughts about the sentiments in the OP of this thread.

Namely, I still stand by the notion that for any deterministic game, humans or a CPU, will eventually solve the game. This happens for humans, in a similar manner, although in much larger time-frames than the CPU to elucidate the winning strategy for themselves.

The downside with this argument is that it is probably near impossible to find two human players at the same equilibrium point to entertain the notion that either side has a more rational player.

Therefore, it seems like we have to constrain the sentiment of two super-rational players that never make mistakes and are infallible with regard to the first move of the player or the response to the first move, et cetera.

With that said, I still believe that humans have the capacity to become super-rational, albeit not in the infallible manner of a CPU opponent. Although, in my opinion this thought is subject to scrutiny after realizing that white, statistically, is a winning player contrary to starting out as black, and that's quite profound in my opinion.

@fdrake, is there any chance you could elucidate what I'm babbling about, and the statement made by @Nagase about statistics being irrelevant, whereas I think it is relevant to elucidating the fact that it is always better for a player to start out as white in chess...

Second, your claim that the first player will "dominate" the game is, again, pure speculation. We don't know that! If you do, I suggest that you write your proof and send it to a reputable journal on Game Theory. — Nagase

I think the statistics of which side wins is sufficient proof. Just that you don't think statistics matters here, which I find puzzling.

I'd be ecstatic if someone could demonstrate the two hyper-rational players would prevent victory from happening for either side, unless a mistake were made for either players. This would apply to games with a finite amount of moves for either player, meaning that the game is deterministic...

A proof is a sequence of statements each of which is justified by appeal to an axiom or to some previously justified (i.e. proved) statement. Statistics are not proof; they are at best heuristics. Otherwise, the Riemann Hypothesis and the Goldbach Conjecture would be considered proved by now. The problem is obvious: even if all the even integers greater than 2 tested so far have been found to be the sum of two primes, that is no guarantee that tomorrow we will not find one that is not a sum of two primes. Similarly, even if statistics show that there is a bias towards white winning, this is very far from a proof that white has a winning strategy. Perhaps this bias is explained by some quirk in human psychology, or perhaps in the next couple of years we will see a reversal towards black winning.