Tuesday, May 3, 2011

Bringing Momos for Others: Game-Theoretic Analysis

Consider the following scenario:

Yet another Sunday evening, the dismal mess food provokes me to undertake long journey from the hostel to SDA- about 2km- on foot to have some momos. The long journey alone is mundane, so looking for company I enquire if my neighbor would like to come along. Rather than being affirmed I meet the reply “Bring one for me as well”, to which I retort agitated not affirming as well and thus he protests “If you're going anyways, what's your problem in bringing one for me? I will pay for my part!” (of course, as if I was supposed to pay for him otherwise). However it's not as if the packet of momos is heavy. So I sit down to introspect whether not bringing is simply sadism. However, it's better to analyze the situation using Game-Theoretic Analysis.

Assign payoffs to the each of the following activities (Subjective analysis):

  1. Walking till SDA: -5

  2. Eating Momos: +4

  3. Eating in mess: 0

  4. Company while walking: +2

By contending myself with the mess food, the payoff would be 0 while by walking all the distance alone I get a payoff of -1 (< 0). For my neighbor, the payoff when I bring him momos is +4 else 0. However my neighbor getting a higher payoff is no justification for me not helping- after all, this isn't a zero-sum game. The following tables would help:

Table 1: The case when I bring him Momos

Payoff(me), Payoff(him)

He goes to SDA

He doesn’t go to SDA

I go to SDA

+1, +1

-1, +4

I don’t go to SDA

+4, -1

0, 0

Table 2: When I refuse to bring him Momos

Payoff(me), Payoff(him)

He goes to SDA

He doesn’t go to SDA

I go to SDA

+1, +1

-1, 0

I don’t go to SDA

0, -1

0, 0

An important observation would be going alone has a lower payoff, for me as well as him. Assuming both players (me and him) to be rational, the natural tendency in both cases is to goof off in the hostel, than walk the distance. So the sentence “If you're going anyways” is wrong. But if both don't go as a result, the payoff for each player would be lower than it would've been had they both gone. An important difference between Table 1 and 2 is “not going to SDA” is no longer the dominant strategy for either player in Table 2. In Table 1, the only Nash Equillibrium is (Don't go to SDA, Don't go to SDA), while Table 2 has two Nash Equilibria- (Don't go to SDA, Don't go to SDA) and (Go to SDA, Go to SDA).

Since the players are rational, in the first case both players would try not going and expecting momos by the virtue of the other's hard work, as a result of which none of them ends up going. Contrary to that in the second case, the rational players would choose between the Nash Equilibria and would reasonably opt for the better Nash Equilibria, so both players would get to eat Momos. An important difference from the standard games studied is here decisions are not made behind hidden doors but rather each player knows the other's decision. Thus they can work out a decision in cooperation. When players decide together, it wouldn't be rational to expect players to arrive at a conclusion where one player gets a higher payoff than the other- both expect equal payoffs.

If one of us goes while the other doesn't, there is a disparity in the payoffs.

  1. The case from Table 1 in such a scenario can be dubbed as exploitation as explained above.

  2. The case from Table 2 is unable to equalize the payoffs, nor raise my payoff. The proposition that one of us goes while the other doesn't means there is a failure in cooperation to go together. So the going player is not reducing the other's payoff, he is just not increasing it (which may sound like a regulatory measure to avoid being taken for granted next time).

  3. The going player would try to equalize the payoffs- i.e. he would assign a cost per unit of payoff the other player gets to benefit from his benevolence, or simply attempt to equalize their payoffs. In the instance of equalizing payoffs, he would charge cost worth 3 units of payoff so that the payoff matrix would now look like:

Table 3: The case when I bring him Momos and charge per unit of payoff

Payoff(me), Payoff(him)

He goes to SDA

He doesn’t go to SDA

I go to SDA

+1, +1

+2, +1

I don’t go to SDA

+1, +2

0, 0

Thus we observe the going player can make profits rather than being exploited using this method. This new situation has 3 Nash Equillibria- (Go to SDA, Go to SDA), (Go to SDA, Don't go to SDA) and (Don't go to SDA). Importantly, none of them are Strict Nash Equilibrium. In such an environment, players may choose Nash Equilibria probabilistically- mixed Nash Equilibrium. In fact, going to SDA is now a dominating strategy for both players. The option where neither of the players go (and thus lose out on momos) is almost ruled out.

The above analysis can be considered as an analogy to exploitation, regulation methods and entrepreneurial incentives. In the last case, we establish the fact that an entrepreneur invests an extra effort where others are reluctant or lazy to do the same, although each of them desires the final output. Thus the passionate and willing ones invest efforts in this direction, benefitting all players in the process at the expense of a meager cost to avoid disparity in payoffs thus eventually making some handy profits.


Ricky said...

I don't know game theory, but I'm fascinated by a thought that in payoff assignment you haven't considered another two factors.

JEALOUSY factor i.e someone getting things for free (I mean without doing any work) is -ve for you. Moreover, y don't u consider the fact that bringing back momos involve ADDITIONAL WORK then walking back alone.

I would be glad to hear your response to this.

Siddharth Bhattacharya said...

like that you can go on with infinite things..like sadism as I mentioned in the beginning...
However, Maths discards emotional relations between people in quantification and thus have been ignored. Here, we speak of objective things- one person maybe jealous while the other may not be. But unlike that, these parameters like company, eating and walking would be universal to all, although the payoffs maybe slightly variant according to the theorist.
A packet of momos is very light- there is no additional work than there would have been otherwise.

Prashant said...

Nice post, attempts to debate were thought provoking, and thats as always the best part!

In amply large cases, a full locality needs pizzas and a dominos outlet opens up.. yes..

but, it would be interesting to analyze without selfish perspective (the key thing), the summation of payoffs positive and negative to say, the whole world.. that happens to be the same as per your post in -1,+4 cases of table 1 in comparison with +1,+2 cases of the last table..
in practice, even in this case, there would be effort overheads associated with charging a cost(negative payoff)..
specially in cases like this overheads wouldn't be negligible..

as a rule i would say, co-operation always results in better summative payoffs than charging, due to overheads associated..

the problems that remain are of disparity in payoffs and motive..

you say charging model, people are driven towards walking upto SDA, and co-operation model, people are driven towards not walking upto SDA..

in practice, people are more mature and human (unlike AI based gamer bots who may have absolutely NO interest in the benefit of others, completely shrewd)..

so, co-operation models do get started with people taking the first initiative..
there should be a feel good payoff factored in also ;)

people imparted with life skills, and existence of social protocols can thus take care of the disparity as well as motive..

exploring another angle ,

again with the aim of maximum summative payoff for everyone..

if i feel like eating momos today, i ask multiple friends (say by posting on the hostel fb group, assuming hypothetically everyone has smartphones) say 7 other people and thus totally 8 people want momos.. if the charging model were to be considered, people would compete (in terms of charging) to go to SDA to get momos, thus mostly more than one person would end up going to the SDA (would be great if you can give a reasoning to determine how many people would end up going, intuitively, i think 4)..

but whats the case for maximum summative payoff, one person going and getting momos for remaining SEVEN people..

here starts the real magic of what co-operation can achieve..

Prashant said...

there may be flaws in what i said though.. i am a beginner here..

Siddharth Bhattacharya said...

@ Prashant:
Thanks for the detailed comment. This post thankfully wasn't an attempt at debate, but rather objectifying the concept using Maths. So let me tell a little about Game Theory in general to you, and its assumptions:
1) Each player is assumed to be rational, that is their sole concern is to maximize their utility function (payoff here).
2) Rationality is common knowledge- Player 1 knows Player 2 is rational, Player 2 knows that "Player 1 knows that Player 2 is rational" and so on... These can be extended for n-player models. 3) Parameters like Goodwill, Selfishness and all subjective things which may or may not be present in a specific human being are overruled here.

If you study some simple games like Prisoner's Dilemma, you will find Cooperating is the best strategy in terms of payoffs, but the 'rational' strategy is to Defect. Cooperative Game Theory and Iterative Game Theory have emerged as a result to help the equilibrium move from defecting to cooperating. Similar is the case here if you analyze Table 1.

In Table 3, in the case of n-player the maximum payoff maybe when one goes and the others don't. However, either all may end up going due to it which converges to cooperation or some will go and bring for the others, while the others agree to pay due to laziness.

So game theory assumes agents to be selfish (or maximizing their own payoff functions), and I have used that fact in writing this post.

Prashant said...

One, I didnt mean that your post was an attempt at debate..

i was talking about my attempts to mentally debate against what you were saying..

yeah i read your previous comment and understood that game theory has certain assumptions and analyzes in a certain way.. fine..

but when we move towards adapting the learnings, and seeing what results in the best outcomes..
the method you proposed was charging for effort..
i appreciate that for being in the direction of considering people's tendencies and leveraging them to effect best outcomes for everyone..

however, i wanted to talk about why it may also be worthwhile for people's tendencies to change.. yeah quite an offbeat thing maybe, and i exactly wanted to demonstrate why it may be worthwhile to still delve into it..

"In Table 3, in the case of n-player the maximum payoff maybe when one goes and the others don't. However, either all may end up going due to it which converges to cooperation or some will go and bring for the others, while the others agree to pay due to laziness."

game theory terms, i thought when N/2 have decided to go, it would be irrational for any more to add to the goers, as he wouldnt see any benefit arising from going (no buyer) and would rather continue as being a customer of one of the goers.. whats wrong ?

and i didnt get the part "converges to co-operation" how would all going be the same as co-operation anyway ?

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