The Optimal Play
December 4, 2009

Ok, so the other day I was playing a 5 player game of Settlers with a few friends. The game got unusually close and heated toward the end, with a 9-6-6-6-6 score. We’ll call the player with 9 points “Alex.” I was sadly one of the sixers tied for second. Alex had just taken Longest Road, which catapulted him from a safe 7 to an unsettling 9 (10 points wins the game). The general consensus was that Alex had the game on his next turn if he was allowed to keep Longest Road. Unfortunately for us, he held the Longest Road by a large margin and we determined that none of us could overcome it alone. After some discussion and plotting, however, we determined that if we all pooled our resources we could grant one of us enough to steal Longest Road (turning the score into 8-7-6-6-6). Alex’s turn had just ended. What would you do?

This is a troubling situation from every player’s perspective. Alex realizes that his best chance at victory lies in us doing nothing. And although we recognize that Alex will likely win if we don’t act, we also recognize that whoever we grant the resources to will be given an enormous advantage and will likely win themselves. This outcome is no better than Alex winning. So what is the optimum play here?

Let’s define what an “optimum play” is (which was what the ensuing argument at our game table was about). Two definitions were considered in the argument: “the play that garners the individual player the greatest chance of victory” (my definition), and “the play that creates the most ‘fun’ and generally minimizes the use of any douchebaggery.” These definitions are each very difficult to assess. How can you measure how an option effects your chance at victory? How can you measure how much fun is generated from your actions? I’ve come to expect that all players in a game should always act in a way that optimizes their odds of victory, but what does “victory” mean?

Let’s switch gears and consider a simple analogous 3 player free for all Starcraft game. Really any multiplayer FFA strategy game will do for this analogy. Player 1 is currently winning, player 2 has about 70% the strength of player 1, and player 3 has about 40% the strength of player 1. Without cooperation players 2 and 3 are essentially doomed, but if they work together they are 10% stronger than player 1. Player 2 should want cooperation the most, as he stands the best chance to come in first once player 1 is eliminated. Now player 3 has little chance at first place, but can choose between second and third (second if he cooperates, third if he doesn’t). The question then is “what is the difference in value that player 3 puts on getting second versus third?” Interestingly, I contend this depends on the particular game and player. In Starcraft, I personally would much rather be second than third. In Settlers, I personally value all non-first places to be about equal (if you’re not first, you’re last).

If the last place players place significantly different values on non-first positions, then we can expect them to behave in a way that maximizes their odds of attaining the highest place they can whereas if they care only for first place they will behave in a way that maximizes their odds of attaining first place. If our Starcraft player 3 cares about non-first positions, then their behavior is easy to predict: they will cooperate with player 2. If they only care about first place, they will behave much more erratically. They might give up hope and just cripple a player based on personal vendettas, they might ally themselves with player 1 and backstab them when they’re weak, they might do the same to player 2 in a desperate attempt at first place. In reality, a player probably experiences some mix of these two victory perspectives, but the mix is probably more or less unique to the specific game and player.

Now that we have defined “victory” I will invoke a concept from basic economics and claim that any “rational” player will act in a way that optimizes their odds of victory. Sure, they could do otherwise. I’ve played with players who get bored of a game and decide to help the winning player just so the game ends faster. I’ve played with players who simply don’t understand the game and make nonsense moves. I’ve played with players who just like “the pretty colors” and make their moves purely on whims. These actions, however, are not rational in the context of the game. In a 6 player game of Monopoly all players could elect to sign over all property and money to a single player and end the game instantly (which in all fairness would probably be the best way to go about playing that horrid game), but do you consider this a “game of Monopoly?” Certainly not. Any reasonable game requires a push and pull of opposing wills struggling for their individual victory. Without that you simply have madness.

So finally, optimal play: “the play that garners the individual player the greatest chance of victory.”

Ok, let’s come full circle to our Settler’s game. If we assume that the players put a reasonable amount of value in non-first positions then we can assume that they would not cooperate to dethrone first place. Why? Because the score would then be 8-7-6-6-6 and dooms the majority to third place or worse. If Alex was allowed to win, then the remaining players (6-6-6-6) would each have a much better shot at second place.

But this was not the case. Apparently, we place high value on first place and very little on even second place in Settler’s. We all conspired to grant one of us (myself as it turned out) nearly every resource available on the table in order to outbuild Alex. This is of course the expected outcome, because if Alex is allowed to take first then it definitively denies it to all other players. Even by widening the 1 point victory gap to 2 points, the majority gives itself just a little more breathing room to get to first.

Sadly we’ll never know how the game would have ended. Everyone felt extremely unhappy about this massive exploitation of the rules and we all gave up after the decision was reached. It’s unfortunate that the nature of the game induced this kind of behavior. An optimal play should never produce such a massive feeling of “brokenness.” This is a failing on the design of the game, and surprisingly I’ve seen this happen many times during my Settler’s career. Tip for game designers: don’t encourage massive acts of collusion. It just makes us upset.