A Game of Tennis Has 20 Possible States, 3 Are Redundant
We recently bought a fake Wii. The real one is too expensive. While playing tennis with my son (I have the sore arms to prove it), I verified something that has been bugging me for about 20 years. The state of “Deuce” is equivalent to “30/30.” I don’t claim to be the first to point this out, but I’m sure it’s true. Turns out there are 20 states in a game of tennis. For those of you unfamiliar with discrete finite automata (DFAs), consider each arrow a play in a game of tennis, and each box a resting point between plays. When player A scores, follow the left arrow and vice versa.
Now, consider if we just removed the state of “30/30″ and went right to Deuce.
Suppose you’re winning 30/15 and you opponent scores. There is no difference in the game if we go to 30/30 or if we just go straight go Deuce.
Just to convince you, let’s combine Deuce and “30/30.” You can see they’re interchangeable.
I just wanted to get this off my chest. It’s been stewing a while.
randyc says:
February 17th, 2010
7:39 pm
Let me offer a long-winded epistemological interpretation: if the true state of the world is 30/30 then from God’s perspective the hypothesis “it’s 30/30″ is strictly better than the hypothesis “it’s deuce” in the sense that one is more true than the other. more true in the sense that one implies 4 serves have occurred and one implies 6 serves have occurred. 30/30 and 40/40 are equivalent in the sense that if A and B (andre agassi and bjorn borg, perhaps?) are amnesiacs and can only rely on the umpire calling game when the game is “truly” over, then the hypotheses “it’s 30/30″ and “it’s 40/40″ are just as good in the sense that neither can ever be falsified, even though one is more “true” than the other. you can imagine other true states and consider the “fit” of the hypothesis. what if the true state is 40/0 and A wins the next point. then the hypothesis “it’s 40/0″ is corroborated, but so too are the hypotheses “it’s 40/30″ and “it’s 40/15″ and “it’s Ad. A”. If A loses and then wins, however, then the hypothesis “it’s Ad. A” is falsified since the ump would call game when A believes it to be Ad A. again. however, the false hypothesis “it’s 40/15″ is again corroborated since the outcome of Lose,Win is the same after states “it’s 40/0″ and “it’s 40/15″ (the outcome is the ump saying “A Wins”). It’s fair to say “it’s 40/15″ is a BETTER hypothesis than “it’s Ad. A” in the sense that it stands up to a greater degree of scrutiny. However, one can not say the hypothesis “it’s one of 20 states” is the BEST hypothesis. It stands up to all outcomes, but not because it’s GOOD, rather because it has no empirical content. That is, it is logically unfalsifiable. If the sequence Lose,Lose,Win is highly unlikely when the true state is 40/0 then the false hypothesis “it’s 40/15″ is still very good in that it will only be falsified with low probability, but at least it’s logically falsifiable. Here’s a conjecture: the hypotheses “it’s 30/30″ and “it’s deuce” (also the pairs 1)”it’s 40-30″ and “it’s Ad. A” and 2) “it’s 30/40″ and “it’s Ad. B”) have exactly the same degree of versimilitude. That is, regardless of what the true state of the world the first hypothesis in each pair stands up to exactly the same tests and fails exactly the same tests as the second. I believe this takes it one small step further than saying ‘if it’s 30/30 and i think it’s 40/40, i’ll never know’, i’m adding ‘if it’s *anything* and i think it’s 30/30 that’s no better or worse than thinking it’s 40/40 (even if neither is “true”)’. You can prove this with brute force, but I’m sure there’s a more “elegant” or (sigh) “sexy” way to do it. It’s not immediately obvious to me that diagram 3 is sufficient.
kapauldo says:
February 18th, 2010
11:50 am
I cannot refute epistemologically because I’m not quite qualified, but I’ll take an amateur’s shot. I suspect you’re saying “there is value in the jump from 30/30 to 40/40 and that value is encoded in the state 40/40.” In other words, I think you’re saying that the states are indeed “useful” because they connote a different quantity of event. I contend that that argument is flawed in the context of my premise. I am merely stating that algorithmically, the states are identical, and that the designers of the game failed to make a different state to encode the value you ascribe. If their intent was to encode such a differential, they would have made the jumpable-to states FROM 30/30 different from those of 40/40. As it is, there is no such value encoded. If gave someone a puzzle that stated, “guess the score of this game? If I get two more points, I win, if my opponent gets two more points, he wins,” etc., and describe in English, every sequence of points from the score you want them to guess, that person could not guess the state (score), because 30/30 and 40/40 are indistinguishable, and therefore encode no differential in value- one or the other
randyc says:
February 18th, 2010
12:17 pm
the “value” encoded in 30/30 that is not encoded in 40/40 has no bearing on “what will happen?” but some bearing on “what has happened?” 30/30 says precisely “it’s tied and 4 serves have occurred” this indicates that the history of the game is one of 6 (i think) paths. 40/40 says “it’s tied and at least 6 serves have occurred” which could indicate one of infinitely many sequences. if encoding historical sequences has value then the inventors should have added scoring rule rather than a finite set of possible scores. that is, they should have allowed 50/50, 60/60, etc. and i agree that if encoding histories has no value then 30/30 can be eliminated. nevertheless, if the score is tied and 4 serves have occurred then “it’s 30/30″ is true and “it’s 40/40″ is false.
this reminds me, have you ever heard of the “red hat problem”? i believe it’s a fundamental nugget of automata theory having to do with “common knowledge” and “interactive epistemology” which are both fundamental to game theory which i teach. but this problem and those of it’s kind are a strange intersection between philosophy and computer science and set theory that can be really addictive. read “on denoting” by bertrand russell.
kapauldo says:
February 18th, 2010
1:39 pm
“nevertheless, if the score is tied and 4 serves have occurred then “it’s 30/30″ is true and “it’s 40/40″ is false. ”
yes, but that’s because of how tennis is encoded, not because they’re not redundant. i.e. your observation is true, but doesn’t affect my proposition. however, it does show that the proposition is ambiguous.
at least we can agree that 30/30 and 40/40 are identical moving forward, and there are multiple ways to get to each, some of those ways are exclusive to one or the other but not both.
strictly speaking, a state machine does not encode history, it only asks “what states can this machine be in?” Were 30/30 part of a loop, as 40/40 is, then that state would not imply that 4 serves have occurred, however, THAT it does, is arbitrary and a result of the encoding as it is. in other words, it’s an artifact.
however, i will have to concede that 30/30 does encode something outside of the state machine, however in the strict interpretation of a state machine, 30/30 and 40/40 are redundant.
randyc says:
February 18th, 2010
2:06 pm
“strictly speaking, a state machine does not encode history, it only asks “what states can this machine be in?” Were 30/30 part of a loop, as 40/40 is, then that state would not imply that 4 serves have occurred, however, THAT it does, is arbitrary and a result of the encoding as it is. in other words, it’s an artifact.” well said. we fully agree. now let me ask this: is my conjecture that ‘the three mentioned hypothesis pairs (and only those three) have the property that the first pair member is empirically indistinguishable from the second pair member from the perspective of any true state’ in some sense a stronger or more general statement? or are they in fact the same? and if it is slightly more general, does figure three constitute a proof in the sense that it constitutes a proof to the redunancy conjecture as you originally intended it?