Goblin Game [From the Archives]

My “Day I Left Pennsylvania” led me to some archived website posts (before blogs were invented) I had written many years ago. I’m re-posting them now. Bear in mind that most of the content in this series is over 5 years old. I have left the content more or less intact. I have removed some links and added some others — but that’s it. Enjoy!


goblin_gameYes it may be a magic card, but it inherently possesses a fundamental of game theory. For those of you who are not familiar with what that image to the right is, it is a card from the game Magic: the Gathering ,however it’s origin is unimportant, nor is whether or not you understand the nuances of the numbers and whatnot. The text in the lower-half of the card is what is important here. Allow me to elucidate in real-life terms….money!:

Let’s say that everyone (3 or more people) has 20 dollars. The game proceeds like this:

  1. Each player hides a certain number of objects (poker chips, for example). The number must be greater than 1.
  2. After everyone has their objects hidden, all players simultaneously reveal their objects to other players. The number of objects hidden is significant here, and should be recorded.
  3. Everyone immediately loses the amount of money equal to the number of objects hidden, this goes into a “pot” in the middle. If a player has hidden more objects than he has money, all of his money is put into the pot instead.
  4. Whoever had set aside the fewest objects loses half of their remaining money, rounded up.
  5. Repeat steps 1 – 4 until only one player is left with any money. That player then wins it all.

Understand how to play? More importantly, do you understand how this illustrates game theory? All players must consider the actions of other players when making their own decisions.

Let’s look at a sample 3-player goblin game:

Player 1
Player 2
Player 3
Wager
Money at
Start of Round
Wager
Money at
Start of Round
Wager

Money at
Start of Round

Round 1
5
20
1
20
10
20
Round 2
5
15
6
9
5
10
Round 3
4
5
2
3
1
2
Round 4
Winner
1
Winner
1
Loss
0

Round 1

In round 1, player 1 takes a conservative 5-unit wager. This is 25% of his total money. Player 2 takes the ultra-conservative minimum wager, betting 1 unit. And player 3 takes an aggressive 50% wager.

The outcome is that players 1 & 3 both wagered more than player 2, so they only lost what they bet. Player 2 is penalized for wagering the least.
Player 1: bets 5, has 15 remaining for round 2.
Player 2: bets 1, penalized 10 for betting the least, has 9 remaining for round 2
Player 3: bets 10, has 10 remaining for round 2.

Round 2

Round 2 sees player 1 betting the same conservative wager of 5, although now it is 33% of his funds. Player 2 has become decidely more aggresive, wagering 66% of his funds with a 6 unit bet. Player 3 is again taking the aggresive stance of a 50% wager, at 5 units.

The outcome of round 2 is that player 2 has wagered the most, and players 1 and 3 are now tied for least, and are penalized.
Player 1: bets 5, penalized 5 for betting least, has 5 remaining for round 3
Player 2: bets 6, has 3 remaining for round 3
Player 3: bets 5, penalized 3 for betting least, has 2 remaining for round 3

Round 3

Round 3 is now the deciding round. Player 1 has already unofficially one at this point, with definitively more betting power than his competitors. Each player bets almost all of his remaining money, saving 1 unit to make sure he doesn’t go broke. Player 3 will lose simply because he cannot possibly bet more than anyone else without betting all of his money.

The outcome of round 3 is as expected: Player 3 bets the least with 1 unit, and is penalized his remaining 1 unit. Players 1 & 2 both end up with 1 unit, and so they will split the money pot, each netting 29 units apiece.
Player 1: bets 4, wins with 1 remaining
Player 2: bets 2, wins with 1 remaining
Player 3: bets 1, penalized 1 for betting least, loses with 0 remaining

Strategy

The strategy behind this game is simple in nature: don’t bet the least. Simplistically, if this were a 1-round game, you would bet all but one of your units every game. However, since a player must plan for the long game, it behooves him to be slightly more conservative with his wagers. He doesn’t have to bet the most every round, but he must certainly not bet the least. The ideal bet is the lowest bet + 1. However that is impossible to do with certainty since so player has perfect knowledge; that is to say, no player knows what the other players are wagering until all bets are locked in.