[Thank you for these comments. My responses are in brackets.]

First, let me state your position as I understand it:

   1. You want to show that a tenable and precise definition of what is and is not a game can be formulated, and you propose such a definition.
   2. You believe that if you fail to establish this, then we are left only with Wittgenstein's "family resemblance" as a way of understanding what is meant by saying that something is a "game" (and presumably a great many attempts to define philosophically important things, such as "knowledge", "intention", etc.).

[No, there may well be better definitions. Or it may be that ‘game’ is synonymous with ‘play.’]

   3. You believe that 2 is "pernicious".

[Well, I think it’s bad for philosophy and unmotivated. Mostly I’m mad at Ludwig.]

My most immediate reaction is that if you convinced Wittgenstein that 1 is true, then he would no longer want to talk about language games. If games are exactly what you say they are, then there would no longer be a point, as far as Wittgenstein is concerned, to remarking that language is something like a game; instead of being an illuminating metaphor, this would merely be a flat taxonomic assertion about language.

[Actually I think my account of games helps illumine part of the work that  LW’s ‘language-game’ metaphor does in his account of language. I write:

'Perhaps the theory sheds some light on the point of
Wittgenstein's treatment of language as a collection of "language
games." The Augustinian account of language acquisition assumes
that human creatures have the natural ability to think about
things, that is, to mentally represent objects in the world, plus
the ability to transfer the contents of thoughts to words. A
language, on this view, is essentially a system of sounds with
derived intentionality, used to communicate information about how
the world stands. As Wittgenstein observes: "Augustine
describes the learning of a human language as if... the child
could already think, only not yet speak." If language is made of
games, however, what counts as success in playing a language game
has no independent or external value or validity; succeeding is
wholly internal to the game, which is what defines success.
Consequently there is no natural intentionality we transfer to
words, the ability to think about things, that determines
linguistic success, the ability to talk about things. Augustine
is mistaken. The meanings of words can be determined only by
their use in a language game. And rules have no derived
intentionality to determine how games are played: the meaning of
a rule is determined by how it is applied, not the reverse. At
bottom, then, intentionality is determined wholly by practices,
what we actually say and do. If language is made of games then
intentionality is determined by behaviour, a consequence from
which Wittgenstein's most striking conclusions flow, for example,
that there can be no private language.']
'
I think that's a good definition of "game". Where I disagree with you is that you think this is the only plausible or accurate definition, and that you (apparently) think that anything that might be called, at some point or another, a "game" and that does not fit this definition is somehow shady, somehow not really a game. I think this implies a mistaken notion of how definitions work in general, and also about the universality of this particular definition.

[No, I don’t think that. I wrote above:

I just want to acknowledge something, in light of several preceding posts (e.g. Gaultiero's). I do agree that we often use 'game' so that it is a synomym of 'play.'
That is, any play is said to be playing a game. What is interesting to me is that, on reflection, we don't say that.

So my trying to stand on my hands, say, to prove my strength or for the recreation, isn't a game.  Trying to balance a stick on my nose to see if I can manage it
isn't playing a game. Throwing a ball back and forth for the exercise isn't playing a game.
Throwing it against the wall and catching it for the exercise isn't playing a game.  A child playing with dolls needn't be playing a game. A child
playing with electric trains isn't playing a game. And so on.

However these are also sometimes said to be games. So I think we use the word 'game' loosely sometimes. Here there is a univocal sense of 'game.' It's what's played.
Anyone playing is playing a game. Not a family resemblance term.

But we also have a more narrow sense of 'game,' which I maintain I've captured. This just gets us a loose and a more narrow sense of the word, not a family resemblance
concept. ]

...
Some games--such as chess--fit your definition very well. Perhaps that's because chess is one of the first examples that will come to a philosopher's mind when he turns to thinking about games. Other activities don't fit your definition well, so you are inclined to skepticism. Mr. Streitfeld advances the case of a child throwing a ball against the wall, and catching it. He is satisfied that this is a game, but you want to analyze the child's criteria for success:

    If the activity is a game, the child counts catching the ball as success because catching the ball is defined as success by an arbitrary rule. She is trying to catch the ball because catching it is defined as success by a rule. If she counts catching the ball as success simply because that’s what she’s trying to do, my definition isn’t satisfied. So the child may or may not be playing a game. Depends why catching the ball is regarded as success.

This strikes me as labored. Perhaps, if we perform an intensive psychological inquisition into the child's ball-throwing behavior, we will get the kind of answer you want. But I think you have then gotten to the point where you are treating the definition as prescriptive: we investigate, and if the behavior doesn't conform to the definition, then we throw it out of consideration.    

[Here’s what I actually write in my paper (quoted in the first post in this thread). I don’t think it’s prescriptive or labored:

Consider the child throwing a ball against a wall and catching it again. Where this activity is a game, the child is following a rule like: "Throw the ball against the wall and catch it, where success is catching the ball you've thrown against the wall." A feature of many children's games is that the activity which is the playing of the game is also what is defined by the rule as succeeding. As the activity is usually easy so is performing it successfully, one of the reasons such games provide so much pleasure for children and so little for adults. The rule for Ring around O' Roses is: "You and your colleagues hold hands and run in a circle chanting 'Ring a round o' Roses.... All fall down'; and all fall down roughly when you sing "All fall down', which is succeeding." Where the very activity that constitutes the game is defined as succeeding, there is success but no winning. Note, however, that in each case we could have adopted different definitions of success consistent with the remaining rules, e.g., "Success is catching the ball twenty times in a row" in the first case, and "Success is being the first (or second, or last) player to reach the ground" in the second.

    There are plenty of recreational activities that are merely pleasant pastimes, not games, e.g., playing with a yo-yo. Here there is no state that counts as success because rules define it. Either we simply enjoy performing the activity, e.g, throwing a football back and forth, or we are trying to do tricks and feats (with the yo-yo, say, or balancing a stick on one's nose) which count as success because we are trying to do them. If a child is bouncing a ball against the wall and catching it simply because she finds the activity pleasant, in exactly the way we find throwing a football back and forth pleasant, she is not playing a game.]

I wonder also about your treatment of boxing. You say that people are reluctant to call it a "game", and you are right. But there are rules of boxing--and not only about where one may strike a blow. There are rules about what constitutes winning a bout. I know almost nothing about boxing, but I know that a knockout (opponent is unconscious on the floor) constitutes a win. So perhaps a contender decides that because she does not have the endurance of her opponent, she will strive to win by achieving a knockout early in the fight. There is nothing personal in this; she beats her opponent into unconsciousness because the rules of the sport count this a success, not because she enjoys inflicting injury or has some grudge against her opponent. So it seems to me that boxing meets your criteria, but you do not want to call it a "game". Why doesn't the definition apply to boxing?

[Here’s what I wrote about boxing:

      This theory of games explains our ambivalence toward calling sports like boxing "games." Imagine the announcer at a prizefight shouting: "He's up, he's down, he's up again! What a terrific game this is, folks!" Yet boxing is included among the Olympic games. Our ambivalence isn't simply because boxing is violent. We have no trouble calling football, rugby, and lacrosse games. Our definition, recall, requires that a game involves a state which counts as success because it is so defined by a rule. Fighting, of course, isn't rule defined. And knocking out your opponent in a brawl is success because that is the goal of a brawl, not because a rule says so. Plainly we made brawling into a sport by crafting the rules of boxing to count as success pretty much what is success in a brawl, rules or no. When we view boxing as a game, we must think that a knockout counts as winning because the rule says so. A knockout would be losing if the rule was different. But we also recognize that the rule defines a knockout as winning the fight because it is winning the fight. Consequently we are ambivalent as to whether the success state counts as success because it is so defined by a rule.]

I want to thank everyone for their comments. I’m involved in two graduate seminars right now and the work load is becoming overwhelming, so I must regretfully leave this discussion. Thanks again to everyone. Much appreciated.