The Games Journal | A Magazine About Boardgames

Basic Strategy 2.0

Greg Aleknevicus

December, 2004

Planning For the Long Term

Many games deal only with short-term optimization—what's the best move you can make this turn? How can you best spend your resources right now? For other games, this might not be the case and poor players will find themselves losing time and again if they take a short-sighted approach. It must be remembered that the object is to have the most victory points (or money or cities or...) at the end of a game, not during it. Knowing when to take a long-term approach is critical for any good game player.

To demonstrate this, let us consider the following fictional game.

Game: Diamond Merchant

Rules: Every turn you are given the opportunity to buy one diamond from a local miner at a cost of $5. At the end of the game, you sell your diamonds to a traveling merchant according to the following schedule:

  • $1 for the first diamond.
  • $2 for the second diamond.
  • $3 for the third diamond.
  • $4 for the fourth diamond.
  • ...and so on.

He will buy as many diamonds as you have to sell. At the end of the game, the player with the most money wins.

Now, consider your first opportunity to buy a diamond—the short-sighted player will decide against it.

"It doesn't make sense to buy a diamond for $5 when I'll only be able to sell it for $1. I'll be losing $4! It's better if I save my money."

In the short-term, this makes sense but it misses an important fact: Buying at a loss now creates the opportunity to buy at a profit later. Since the short-sighted player never buys a diamond, he's always faced with the same decision: buy a diamond for $5 and sell it for $1.

Clever Hans, always a forward thinker, realizes that by taking an early loss he can (hopefully) make a profit later. He notes that the fifth diamond he buys will actually cost him nothing. Even better, the sixth will earn him a $1 profit, the seventh $2 and so on.

So as long as Clever Hans can buy six diamonds, he'll be able to make a profit, correct? Not so fast! It's true that he'll make a profit on the sixth (and subsequent) diamond purchased but (being clever) he's more concerned with his overall profit. In other words, he needs to account for the losses he incurred buying those first five diamonds. Doing a little math he quickly determines that he will need to buy nine diamonds to break even:

Cost: 9 diamonds at $5/each  =  $45.

Income: $1+$2+$3+$4+$5+$6+$7+$8+$9  =  $45.

So, if Clever Hans can buy nine diamonds over the course of the game, he'll be no worse off than if he had purchased none. The big advantage is that with every additional purchase, he'll make an overall profit and this profit will increase with each purchase.

Cost: 10 diamonds at $5/each  =  $50.

Income: $1+$2+$3+$4+$5+$6+$7+$8+$9+$10  =  $55.

This is illustrated in the following graph:

Also note that the level of profit rises rapidly with each purchase: buying 10 diamonds nets a profit of only $5 but purchasing 13 nets a profit of $26!

The critical evaluation is determining if you will, over the course of a game, have enough opportunities to take advantage of your long-term strategy.

In the case of Diamond Merchant, this is simple—if the game lasts ten or more turns, Clever Hans' long-term strategy will pay off; if it lasts eight or fewer turns, it won't. It will be far more difficult to determine the number of "buying opportunities" in a real game.

Complications

One of the reasons why Diamond Merchant is so easy to analyse is that the cost and payoff are the same unit: dollars. In most real games this will not be the case and so you will often have to purchase "diamonds" with dollars whereas you get paid in "victory points". This comparison of apples to oranges is critical and can make much of the above analysis flawed. For instance, if money is worthless at the end of the game and cannot be used for any other purpose, then buying even one diamond makes sense—1 VP is worth more than $5. Luckily, the information is not totally useless as most games offer you a variety of things to purchase with your cash. The question then becomes one of "opportunity cost".

Let's add a couple of rules...

Game: Diamond Merchant 2

Rules: Every turn you are given the opportunity to buy one diamond from a local miner at a cost of $5. Alternately, you may instead buy one ruby, also at a cost of $5. At the end of the game, you sell your gems to a traveling merchant and he pays according to the following schedule:

  • 5 VPs for each ruby.
  • 1 VP for your first diamond.
  • 2 VPs for your second diamond.
  • 3 VPs for your third diamond.
  • 4 VPs for your fourth diamond.
  • ...and so on.

He will buy as many gems as you have to sell. At the end of the game the player with the most VPs wins.

Now things are a little trickier. Since money is worthless at the end, it makes sense to purchase gems even if you can buy only one. The question becomes a choice between rubies or diamonds. The short-sighted player will decide on rubies because they pay 5 VPs whereas your first diamond pays only 1 VP. Clever Hans takes a more considered approach and asks himself how many gems he's likely to purchase over the course of the game. After all, rubies are worth more only compared against the first four diamonds purchased. The sixth, and subsequent, diamonds are worth more than rubies. As in the previous problem, the question is whether you can make sufficient profit on later purchases to overcome the money "lost" on earlier ones. The following graph shows the total profit of buying diamonds and rubies.

As can be seen, the "break even" point is at nine gems at which point it makes no difference whether you have purchased all diamonds or all rubies, your total profit is 45 VPs. However, if you have the opportunity to buy only eight or fewer gems then rubies are your best choice whereas if you can buy ten or more, you're better off with diamonds.

Practical

In Reiner Knizia's Taj Mahal, you gain points in a variety of ways but many people consider the elephants (which grant you commodities) to be very powerful. At first glance this might seem counter-intuitive. You don't get to place a palace for winning the elephants and so miss out on those points as well as missing out on any of the bonus tiles. So why are they so desirable? In Taj Mahal, you receive one VP for every commodity you win plus one VP for every one of that type you already own. Your first box of tea gives you one VP, your second gives you two, your third three and so on. (This is the same payout structure as our fictitious Diamond Merchant.) Players who fail to recognize the power of this are usually in for a surprise as the game continues. Pondering the current game state, they'll see that if they win the current commodity tile they'll gain three or four points but if an opponent wins it, she'll gain nine or ten! This is not a particularly happy revelation as the game draws to a close.

Note that the positioning of palaces (and the order that provinces are decided) also rewards the forward-thinking player in Taj Mahal. While you gain only a single VP for winning one or more palaces in a particular province, you also gain points for every province in which you create a chain of palaces. It's this reason why it's often in a player's best interest to place a palace on a "blank" spot rather than one which grants a bonus tile—fewer points right now can lead to greater points later.

- Greg Aleknevicus

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