Reading this book as a game designer is both fascinating and a little odd. Chapters 3 and 4 have both taken the same basic approach: analyzing the best way to "solve" a specific type of game.
My knee jerk reaction to this approach is concern. If you tell a game designer that you solved their game I suspect you are breaking their heart. It would certainly break mine. To me, solved games are broken games.
But I think designers can take this a step further. We can't stick our heads in the sand and pretend our games aren't broken. Instead we should use the tools Dixit, Skeath, and Reiley discuss to discover the specific bits that are broken so they can be fixed.
So what are we looking for? In chapters 3 and 4 the authors have focused on finding three different things that would make your game broken: Dominating Strategies -- the strategy that always wins, Dominated Strategies -- the strategies that never win, and Nash Equilibria.
In the hobby game environment a Nash Equilibrium feel to me like those odd ruts you get into where Player A feels obliged to do X and Player B feels obliged to do Z in response and they never really have a good reason to do anything different. In fact, every other action they could take seems less optimal. I suppose this is just a redefinition of a Nash Equilibrium -- a combination of strategies or choices where each player gets the best value given the variety of each others' responses.
And how do we find them? In chapter 3 the authors discussed decision trees. In this chapter they introduce the decision matrix. Where trees helped analyze sequential move games (I take a turn, you take a turn, ...) matrix help analyze simultaneous move games.
Above is a matrix for Rock-Paper-Scissors. The results are expressed from the perspective of the Right Hand. RPS is a 0-sum game so we only need one set of results. In non-zero-sum games you would list a number for each player showing their reward in each situation.
One thing you can say about RPS is that it does not have a dominant or dominated strategy. Below we change the rules of the game to say that rock always wins but still ties against another rock. And the matrix clearly shows that rock is the dominant strategy.
Now this would have been obvious without a matrix, but most games are more complicated. Below is a decision matrix for a very specific situation that can come up in Championship Formula Racing.
Image two cars in a racing game sprinting for the checkered flag. The trailing car could end up gaining spaces from slipping the lead car in certain situations. Without going into all the rules and reasons why, here are the cars' choices and results from the perspective of the lead car. Also note that a result of 0 will likely mean that the lead car wins.
Dominated Strategies: For the trailing car, going 160 is dominated... it never results in the trailing car winning.
Dominating Strategies: There are no strategies for either car that always wins. 200 is closest for both cars, but if they both play 200, the trail car will win.
Nash Equilibrium: There is no Nash Equilibrium either. No matter what speed one car picks, they might have a better play depending on what speed the other car picks... basically this scenario turns into a guessing game.
Below is a matrix for the same situation but with different choices.
Dominated Strategies: If the trailing car goes either 140 or 160, they can't win (remember 0s mean they will lose).
Dominated Strategies: Meanwhile the lead car can not lose if they go 160.
So this situation is clearly solved. If this where the whole game, this would be a problem I would need to fix.