Game Theory: Play or get played?

"I'll go first", "Oh, come on it's my turn this time." (every game is like this isn't it?) Most of the time its a coin flip that decides who goes first and of course you believe it to be 'fair'. So when your friend places a coin on his thumb and says "call it in the air", you realize that it doesn't really matter whether you pick heads or tails.

Every person has a preference, for sure—heads or tails might feel "luckier" to you—but logically the chances are equal, or are they? 

The 50-50 proposition is actually more of a 51-49 proposition, if not worse.  51% chance of landing on the same face it was launched. (If it starts out as heads, there's a 51% chance it will end as heads) The sacred coin flip exhibits (at minimum) a whopping 1% bias, and possibly much more (What, even this isn't fair now?)

Games that tend to be a fair deal, are actually not (We just saw how we lost those bets cause of the unjust coin flip) To talk more about games, we must know what Game theory is. (Yes you heard it right, we now have a theory on games too)

Game theory is the study of games (its a lot more than ludo king). Game is a situation having multiple participants and in which your action is dependent upon the action of other players. You'll only decide what's best after they decide (peer pressure?) So chess is a game and so is poker (and what about ludo king) But sudoku and crosswords are not since you solve them independently sitting on your sofas without being affected by other's actions.

Here are some situations that you'll be amazed to know and how game theory actually realtes to them. 

PRISONER'S DILEMMA

You were involved in a money heist with your friend. (no it was not the Royal Mint of Spain). The police caught you later but they don’t have the proper evidence to give you both the full punishment. They give you the following options-

If you confess (Did you confess to him yet?) and your friend confesses you both will get only 5 years in prison. If you both remain silent (the way when the professor asks you a question) you both will get only 3 years of prison. If you confess (traitor) and your friend doesn’t, you’ll go free and your friend will get 10 years of prison and vice versa.

What should you do? Let’s study this payoff matrix to analyze what would be the best situation for you (and your friend?)

Studying this matrix, it is quite clear that you and your friend both will be better off staying silent. (Shhh) But will you?

This is where game theory comes into play (you were waiting for this, weren’t you?) The Game Theory combines human emotions and the probability of events to determine what is the most likely outcome. You’ll decide whether you’ll confess or stay silent based on-

                                                            

 

If you know your friend will stay silent, you might think confessing would be better since you’ll serve no jail time (can plan the next heist maybe?). If you think this who’s to say your friend won’t. In situations like these, trust is a difficult thing (we all are familiar with “trust issues”). Chances are you both will confess and serve 3 years in jail (together). This would be a stage of Nash equilibrium. (what?)

Nash Equilibrium is a pair of strategies in which each player’s strategy is the best response to the other player’s strategy. In a game like the Prisoner’s Dilemma, there is one pure Nash Equilibrium where both players will choose to confess. However, the players only have two choices: to confess or not to confess. What happens if there are more choices? 

ROCK, PAPER, SCISSORS

Remember stone paper scissors that we all used to play in our old school classrooms whenever we got a free period or got bored of studying. In this classic game of rock, paper, and scissors, there are three choices-losing a round of the game results in a payoff of -1, winning a round results in a payoff of 1, and tying in round results in a payoff of 0. If player 1 plays rock and other players too, it results in 0 points for both. However, if a player plays paper but the other one plays rock, he gets +1 but the other wins and hence gets -1.

                          

 From the payoff table for the game of rock, paper, scissors, it becomes evident that there is no such equilibrium. There is no option in which both players’ options are the best response to the other player’s option. Thus, there are no pure strategy Nash equilibria. It is thus a game of mixed strategy.

Coming to the most interesting part, is there any way to win this game? Yes, the game we thought is purely based on luck does have some strategies.

The first strategy is countertactic. Let's say you played scissors and your opponent played rock. The chance that your opponent will confidently play rock again is now very high. Thus, you need to play the option that wasn't played in the previous round. The second strategy is to mirror. If you just won, play what your opponent just played, because he or she will think that you are going to play the same gesture again. This elaborate tactic lies at the intersection of math and psychology. It's based on game theory, the science behind how humans make decisions in competitive situations. (I am sure all of us are dreaming of participating in the world championship of rock-paper-scissors now)

MEXICAN STANDOFF (Sherlock, Moriarty Face Off)

Now let’s talk about Sherlock (yes, we are talking about our favorite high functioning sociopath). Once he was involved in a Mexican face off with Moriarty (miss me?) and a sniper. Now the sniper is well hidden and is not any danger but its perfectly clear with the gun pointed at their heads that who’s in complete danger. What happens next? As always sherlock manages to surprise us. He points his gun at the explosive in the room. Now the sniper is in danger too. In fact, everyone is in equal danger now which diffuses the situation. This is again an example of game theory since the action of one person is dependent on that of the other person (the same way our assignment is dependent on our friend completing that assignment)

NOW IT'S YOUR TURN

Coming to the last problem of the day. This one's for you to solve. Imagine you (A) and 2 of your other classmates, let's call them B and C have to go home after school. The bus was not available that day you had to take a taxi. You 3 live in the same street but your houses are at a distance from each other such that- If you all take a taxi separately it costs Rs 8 for A, Rs 12 for B and Rs. 40 for C. If you go together it'll cost Rs 40 total as A and B get off earlier and only the distance of C will count. What

will you do? Should A (you) alone pay or together and if together how should you split the bill? Equally?  Proportionately? or are you smart enough to think of a new way xD Let us know in the comments! (Don't google guys)