Heads I win, Tails you lose.
June 28, 2010 1 Comment
For the past few days, I’ve been reading the lovely book, The Art of Strategy, that I wrote a couple of years back, when I was in college. Okay, I was bluffing. Avinash Dixit and Barry Nalebuff wrote it. And, in case you were really credulous enough (euphemism for dumb?) to think that I indeed authored a book in my undergrad, then this book is probably just for you. You would probably need all the strategic thinking practice that the authors give you. Steven Levitt, coauthor of the celebrated Freakonomics, said about this book: “I liked it so much, I read it twice.” 🙂
Very few books are educative and fun at the same time. This one is! The authors take us on a thrilling ride exposing the fascinating facets of game theory.
Okay, let’s play a game that the authors designed for us. Whenever we need a impartial referee to resolve any disputes, we shall bring in the irrepressible George W. Bush. You see, if the referee is a strategic thinker himself, he might take sides….
So we are in a big open grass field. George W. Bush has planted 20 flags in the field. Mr. Bush was also kind enough to tell us the rules. You go first and remove either 1, 2 or 3 flags. Then I’ll go and remove 1, 2 or 3 flags. Then it’s your turn again to remove flags, followed by my turn, and so on. You win if I remove the last flag. I win if you remove the last flag.
(Wait a sec. Mr. Bush has something to say. “Both of you have to remove at least one flag whenever it’s your turn. You can’t say that you don’t want to remove any flag.” …. Nice rule that one.. If there was only one flag left, and it were my turn, my devious plan was to run away. That way, you wouldn’t have won. Mr. Bush…prescient?? Wow!)
Bush: Gentlemen, let the play begin! (To evade accusations of gender bias, I promise that if I ever post on game theory again, you will be assumed to be a girl. )
You remove 3 flags.
Ok.. 17 left…
I remove 1 flag.
You remove 2.
14 left ….
I remove 1.
(Ah… there’s a “I know something that you don’t” sort of grin on my face now. 🙂
You ignore my facial expressions and remove 2 flags.
11 left …
I remove 2.
9 left …
You remove 1.
8 left …
I remove 3.
5 left …
Oh wait. You suddenly seem to become dispirited. Come on, it’s just a game!
Putting on a brave face, you remove 2 flags.
3 left …
I remove 2.
1 left …
So was this a pure game of chance? Na…
Let’s assume there were only 2 flags (instead of 20). Also, let us assume I take first turn. Okay, so I remove one. You’re doomed.
Now, let’s assume we had 3 flags planted in the ground. Again, me first. I remove 2 flags. You are left with the last flag. You’re again doomed.
Okay, now let’s assume there are 4 flags on the ground. My turn again. I remove 3. Sorry. 🙂
So, we notice one thing here. If it is my turn, and there are either 2 or 3 or 4 flags on the ground, I can always play cleverly and ensure that I win.
Now, let’s assume there are 5 flags on the ground. And it is my turn. If I remove 1 flag, you are left with 4 flags on the ground on your turn. Therefore, you can apply what we saw above, and ensure that you win.
So, if I remove 1 flag, you can ensure that I lose.
What if I remove 2 flags? You are left with 3 flags on your turn. Again, if you are clever, you can ensure that you win.
Now, what if I remove 3 flags? This time you’re left with 2 flags. And once again, you can ensure that I lose.
So what do we observe here? If there are 5 flags on the ground and it my turn, then I am doomed (assuming you are smart).
To summarize what we saw so far … (Let the two players be Player A and Player B.)
– If it is Player A’s turn and there are 2, 3 or 4 flags on the ground, then he can always ensure that he wins. (In other words, Player A has a “winning strategy” [ a strategy using which Player A will win irrespective of what Player B does]. )
– If it is Player A’s turn and there are 5 flags on the ground, then whatever Player A does, he can not win (assuming, of course, that Player B is clever).
Returning to our game, my strategy will be to ensure that when it is your turn you are left with 5 flags. And, also that, by no chance, should I leave you with 2, 3 or 4 flags on your turn.
Okay, now let’s suppose there are 6 flags on the ground, and it is my turn. What do I do? I remove 1 flag. That leaves you with 5 flags, which, as we just saw, places you comfortably on the “highway to hell” (I just wanted to include that phrase; I know it doesn’t quite fit in here. Ah who cares. 🙂
So by removing 1 flag from 6 flags when it is my turn, I can ensure that I win. That is, just to re-emphasize (at the cost of repetition), my winning strategy when I am left with 6 flags, is to remove 1 flag.
What if I have 7 flags left on my turn? I remove 2 flags, of course. This ensures that I will win eventually (i.e. winning strategy).
8 flags, my turn? I remove 3 flags; placing me comfortably on the “stairway to heaven”. (Again, just wanted to use the phrase. )
9 flags, my turn? I am doomed. Whatever I do, you can ensure that I lose.
Thus, if Player A and Player B are in this game, Player A’s strategy will be to leave Player B with 1 or 5 or 9 flags. Player B will have a corresponding strategy with just the roles reversed.
You see the pattern? If I leave you with 13 flags, you will necessarily lose. Similarly, if you leave me with 17 flags, you can make sure that I lose.
So, when you started the game with 20 flags, had you removed 3 flags, and played sensibly thereafter, you would have ensured that you win.
This is just one of the numerous interesting games described in Dixit and Nalebuff’s book (and a very simple one at that).
Unable to find a fine ending to this post, I think I’ll just copy what Dixit and Nalebuff wrote at the end of the Introduction:
“We warn you that some of the strategies that are good for achieving these goals may not earn you the love of your rivals. If you want to be fair, tell them about our book.”