Ice Cream Theory
Updated: Nov 21, 2019
I-Scream, You-Scream, We-All-Scream
Ice Cream, the divine food of the gods that was somehow discovered by man and then CONSUMED (almost every day, haha) by me. We all have, at least, one tub of ice cream in the fridge that is our go-to after a long day at “Uni” or work.
Figure 1. So true
The problem is that everybody in that house is tired after their dreary day of drudgery *suspiciously looks around the house* (cue intense music) but YOU NEED TO GET YOUR HANDS ON THAT TUB FIRST! Enter Game Theory, a concept that was conceived in a bar in order to woo girls by Robert Nash (YES, it’s the same “Nash Equilibrium guy!), which will now be employed by you to get your ice cream! I am also aware that you must be wondering how a matrix with some words and numbers will help you get your ice cream but trust me there’s more to game theory than meets the eye(scream).
“A Matrix with Numbers and Words”
I hate Math and I can precisely pinpoint when that happened: Algebra! The moment they introduced letters of the alphabet into Math was the moment my brain went, “Aren’t Math and English two different things?” “They are other Dhruv, they are.” (that’s an Archer reference for all the people who didn’t get it) The point of my sorrowful relationship with Math is that, eventually, when it was taught to me in a fun and relatable manner, I was able to hate it a little less (a small marginal amount) and I’m hoping that I can do the same for you. Maybe after this cheat sheet, this “Matrix with Numbers and Words” will have a different impact on you when you see it in the exam, your notes or anywhere!
Figure 2. The next time the cookies go missing, you know who to blame
Ice Cream Theory 101
*Comes into class, puts papers down, tests the mic, clears throat* “Let the GAMES begin!” “Hold on! What about the rules?” Well, the rules are pretty simple: we, each, decide what to do “simultaneously” (hence, the annoying “Simultaneous Game”) and then execute. Here’s the catch, the both of you all can’t communicate with each other, you just have to choose one option and go with it!Dhruv/ShreyanshDon’t EatEatDon’t Eat5,50,10Eat
Table 1. It’s time to go to war
From the table above we can conclude that the optimal… (Umm, I don’t understand what he means, ugh) Let’s take a step back, shall we? The table is color coded where all the orange (or, I think its orange) numbers represent my (Dhruv’s) payoffs or happiness from eating ice cream and the blue numbers represent the ice creamy-goodness that Shreyansh, my ice cream stealing roommate (always finishes that amazing Belgian Chocolate before I can get my hands on it, but not this time!) gets from consuming them. Here comes the tough part, you guys know that if Shreyansh doesn’t eat the ice cream and I do (God, I hope that this always happens) then I get all the happiness (as represented in the bottom left corner of the matrix) and that he gets all the happiness if the opposite occurs (as represented in the top right corner of the matrix). What do the “5,5” and the “1, 1” mean though? This is when those seemingly confusing “Optimal Solution” and “Nash Equilibrium” come into play. It’s common sense to conclude that the best option for, both, Shreyansh and I is to cease all enmity, come to compromise and share that ice cream whereby we both get 5 units of satisfaction: it’s a win-win. We, however, conveniently disregard our “Selfish greed” (or at least that’s what Economists would have you think) because after all, we are “rational” individuals and who in their rational minds would want to share the ice cream when you can have it all for yourself!
You (assume that you are me, Dhruv, in this case) have decided that you’re tired of letting Shreyansh eat the ice cream all the time, you’re tired of getting the shorter end of the stick and so, you decide to use your giant brain (hehehe) to come up with a solution that guarantees you the ice cream, every single time. You start thinking, “What if I can predict what he does? I can then, accordingly, choose the best option.” That sounds about right, so now you have to think of what Shreyansh could possibly do and it boils down to two simple options: he either chooses to eat the ice cream or chooses not to (although its never going to happen because he’s an ice cream guzzling monster, we want to cover all our options though, right?).
Dhruv/ShreyanshDon’t EatEatDon’t Eat5,
1ST HALF 0,
2ND HALF Eat
1ST HALF 1,
Table 2. Only Dhruv’s Payoffs
The most likely scenario is that he WILL chose to EAT the ice cream and so, when you weigh your options, you find yourself in the 2nd HALF. Given that “Eat” is what Shreyansh is going to do, a satisfaction of 1 is still better than 0 and so you’re gonna fight him to the death for that ice cream. If by some divine stroke of luck, where God Himself wants you to eat that ice cream, Shreyansh doesn’t feel up to it (you’re therefore in the 1st HALF), then eating the ice cream and the subsequent satisfaction of 10 is better than not eating it (where you get a satisfaction of 5). Here’s the interesting observation though, regardless of whether Shreyansh eats or doesn’t eat the ice cream, your DOMINANT STRATEGY is to eat the ice cream! Go back and read the strategy we just discussed and you’ll see that your best option to get the most satisfaction from this encounter is to always try to eat the ice cream, regardless of what Shreyansh does! Congratulations! You have got your plan all sorted out! With elated spirits, you run to the fridge only to find that Shreyansh is ALREADY THERE! It is futile to talk about what ensued but all I can say is that, “At least he didn’t get all the ice cream.” (because most of it is on the floor post the “battle”).
The Sub-Optimal Solution
Why didn’t our strategy work! Is it because Shreyansh thought the same thing? While it is very nice to disregard Shreyansh’s intellectual capacity, it’s very likely that he’s thought of something like this before. After all, he has been eating almost all the ice cream so far, right? This is what he was probably thinking when he charted out his options:
Dhruv/ShreyanshDon’t EatEatDon’t Eat,5
1ST HALF ,10
1ST HALF Eat
2ND HALF , 1
Table 3. Only Shreyansh’s Payoffs
If you think about in the same way that we did, you’ll conclude that Shreyansh, too, has a dominant strategy: he must eat all the ice cream that he can get his hands on! Keeping this in mind, it’s all well and good to have a strategy that takes into account how your rival/opponent thinks, the end result is what truly matters. If, both, Shreyansh and I decide to always eat the ice cream, we’ll end up in the infinite loop of the “Eat-Eat” corner and always gain a of satisfaction of “1,1”. The technical term to explain this state of equilibrium is “Nash Equilibrium”. It is defined as “a stable state of a system involving the interaction of different participants, in which no participant can gain by a unilateral change of strategy if the strategies of the others remain unchanged.” What that means in, (much needed) simpler terms is that it is when the people in a game have no reason to change what they’re doing, given that the other players/people in the game do not change what they are doing. A Nash Equilibrium is a state of equilibrium and many such equilibria can exist in one game itself, you just have to incentivize the players with the appropriate satisfaction or “payoffs” to sustain this (don’t worry, I’m not going to bore/confuse you with this). In our game, there are two such equilibria: one where Shreyansh and me communicate with each other and come to a consensus of sharing, peace and prosperity (and the subsequent satisfaction/payoff of 5,5) or the one where we don’t and a chaotic battle ensues with great loss of life, friendship and (more importantly) ice cream!
Yes, yes. We’ve Come to the End
There’s so much more to Game Theory than what’s in our textbooks or notes! If you can apply it to something as simple as ice cream, imagine the widespread application! I hope that by the end of this that you’ve been able to understand at least a little bit of game theory and don’t view it as an annoying part of an even more annoying “Econs” module. In fact, we didn’t talk about the strategies of players over repeated games and how we can sustain any Nash Equilibrium with the right incentivization. On a more a serious note, this concept of cheat sheets is very new! If you have any ideas about further cheat sheets, don’t hesitate to get in touch with us! Anyway, I will take my leave now *disappears amid a cloud of smoke*
Google Dictionary (Online); https://www.google.com.sg/search?q=definition+of+nash+equilibrium&oq=definiton+of+nash&aqs=chrome.1.69i57j0l5.5902j0j9&sourceid=chrome&ie=UTF-8 Accessed on 2nd October, 2018