The tables are shown only for the initial moves. But I agree that the number of game states is enormous (e.g [ 6!6!C(13,6)^3 ] after 6 plays ?) so I’d like to understand his heuristic shortcut.
Goofspiel could be scored at $1 per point, which would simplify things – the prior scoring would be irrelevant if not the cards played. But Rhoads solves for the all-or-none game.
If you’re talking in terms of a win or a loss, how can you say “players don’t care about anyone’s score but their own”?
A player cares about “do I win or do I lose”. The player does not care about “does my opponent win or lose”. Though of course, many games are zero-sum, meaning that the answer to one question will completely determine the answer to the other.
That’s my point. In a game in which there is a winner and a loser, how can somebody meaningfully say that “I don’t care whether or not you lose, as long as I win”?
Exactly like that. Their behavior in zero-sum game is identical to that of someone who wants to minimize their opponent’s payoff, but that doesn’t mean that they’re doing anything but maximizing their own expected payoff.
If you really do care about keeping the other guy’s winnings low, I can design a game where I’ll take as much money from you as I want.
No, that’s just the other side of the flawed “score high” strategy.
You can lose a game with a strategy that maximizes your own score. And you can lose a game with a strategy that minimizes your opponent’s score. But if your strategy scores more points than your opponent, then you’re always going to win.
Points have nothing to do with it. As septimus said, the payoff in a baseball game is essentially $1 to the winning team, and $1 from the losing team. That’s what you’re trying to maximize, not the number of runs.
Your score is not necessarily the same thing as your payoff. Payoff is the same whether you win by a little or by a lot.
But sometimes score and payoff are the same. If they are, then it may occasionally be true that the straightforward score maximization won’t win you the game. Corner solutions occasionally happen when the payoff space is somehow constrained. So a rational player checks critical points in the utility function to see if any of them do better than straightforward maximization.
Or rather than maximizing your score and minimizing your opponent’s, you just maximize the nonnegative difference between scores.
Then you can’t maximize it at all. Every win and every loss is worth exactly one dollar.
Now if you’re talking about the course of a season and who goes to the playoffs, then you want to win more games or you want the other teams to lose more games - which means you’re back to points with each game counting as a point.
Maximizing the difference between scores is a good strategy but it’s not the best one. The best strategy is one that gives you a higher score than your opponent.
Here’s an example of the difference: two players are rolling dice and whoever has the higher roll wins. Now a referee offers to give you an advantage in the game and lets you pick from two choices. The first choice is that you both roll your die and you add one to whatever your roll is and your opponent subtracts one from whatever his roll is. The second choice is that you don’t roll a die; your opponent rolls his die and then you get a number that’s one higher than his roll.
Now mathematically, the first choice will make your rolls, on average, two higher than your opponent’s and the second choice will make your rolls one higher than his. So if you’re looking to maximize the nonnegative difference between scores, you’d choose the first choice. But if you want to guarantee you’ll score higher than your opponent, you take the second choice.
The best strategy is the best response to the best response of the other players in the game. Like everyone has been trying to say, this depends on what you actually take home when the game ends. In an auction game, it doesn’t just matter whether you win the auction but how much you end up paying for the auctioned item. In a game of chess, it doesn’t matter whether you have two pieces left or twenty, as long as you win. What quantity you maximize obviously depends on the payoff of the game.
The game you propose is a non sequitur because the game isn’t even strategic. Your opponent has no turn at all, so your best strategy is not a best response to any play your opponent makes. Not winning all the time is always weakly dominated by winning all the time, unless the payoffs from winning occasionally balanced by losing occasionally are greater as the game is repeated than taking a winning payoff in every turn.
My point is that the claim that “in standard game theory, players don’t care about anyone’s score but their own” is not accurate. There are numerous situations in game theory where a player needs to care about what scores other people have.
Only in so far as the scores of others influence their own score.
I was trying to avoid jargon when I said that, but it’s probably not that great to have the same word “score” meaning two different things. The accurate claim is that rational agents act to maximize their own expected utility without regard to any other agent’s utility, but you can see why I thought that might be a little more difficulty to understand.