BrettRants

Tit for Tat Logical Fallacy


Classically, Tit for Tat is a math problem
With different numbers, different conclusions may be drawn.

Tit for Tat is a mathematical problem in game theory in which the outcome for each turn is dependent upon how a player and their opponent move.

Each player (Player One & Player Two) decide how to play (Nice or Mean) and they get points dependent upon how the other plays.

The game is called Tit for Tat because the computer program that won this first program had two simple rules:
  1. Play 'Nice' to start
  2. Play whatever the other guy just played on your next turn ('Nice' for 'Nice' and 'Mean' for 'Mean', otherwise known as Tit for Tat)
Here's a sample payoff chart (Payoff Chart #1) as was used in the original game (or close enough for our purposes).  

Payoff Table #1Player Two:NicePlayer Two: Mean
Player One: NicePlayer One: 3, Player Two: 3Player One: 0, Player Two: 5
Player One: MeanPlayer One: 5, Player Two: 0Player One: 1, Player Two: 1

Rather than doing any number crunching, I'll simply point out that Payoff Chart #1 was arbitrarily decided upon by the game's creator.
So, we could play a slightly different game with a different arbitrary payoff chart (see Payoff Chart #2) and maybe get different results.:

Payoff Table #2Player Two:NicePlayer Two: Mean
Player One: NicePlayer One: 9, Player Two: 9Player One: 0, Player Two: 0
Player One: MeanPlayer One: 0, Player Two: 0Player One: 0, Player Two: 0

Of course, we don't get different results.  If using Payoff Chart #1 the best strategy for both players is to play 'Nice'.  And this remains the best strategy is using Payoff Chart #2 (only hopefully it's a lot easier to see) as playing 'Nice' is the only way for anyont to get any points.

Here's another chart (Payoff Table #3) that makes a different winning strategy easy to see.

Payoff Table #3Player Two:NicePlayer Two: Mean
Player One: NicePlayer One: 0, Player Two: 0Player One: 0, Player Two: 0
Player One: MeanPlayer One: 0, Player Two: 0Player One: 9, Player Two: 9

Hopefully, it's pretty much a no-brainer to play 'Mean' if using Payoff Table #3.
(And here we perhaps get into the metaphysics of 'Nice' and 'Mean' and realize that mathematically they mean no more than X or Y).

Anyhow, things get interesting if we change the chart again (per Payoff Table #4):

Payoff Table #4Player Two:NicePlayer Two: Mean
Player One: NicePlayer One: 0, Player Two: 0Player One: 0, Player Two: 5
Player One: MeanPlayer One: 5, Player Two: 0Player One: 0, Player Two: 0

Payoff Table #4 is interesting because the best move for each player is to play 'Mean', but they don't get any points if the other doesn't play 'Nice' to their 'Mean'.  Or in other words, the mutual winning strategy is for one to play 'Nice' to the other's 'Mean' and then switch roles.

We could further refine the table as per Payoff Table #5:

Payoff Table #5Player Two:NicePlayer Two: Mean
Player One: NicePlayer One: 1, Player Two: 2Player One: 0, Player Two: 4
Player One: MeanPlayer One: 8, Player Two: 0Player One: 2, Player Two: 1

To figure out the 'winning' strategy without crunchin any numbes, all we have to do is look at the payoff chart and figure out how Player One scores the most and then do the same thing for Player Two.  In each case (because I designed the chart that way), each player scores the most if they play 'Mean' to their opponent's 'Nice'.  So, each would score the highest if they alternated.  Or (and this is where it gets really interesting), Player One would wind up with the same score as Player Two if they played thusly:


Turn NumberPlayer OnePlayer Two
1Mean: 8Nice: 0
2Nice: 0Mean: 4
3Nice: 0Mean: 4
4Nice: 0Mean: 4
5Nice: 0Mean: 4
6Mean: 8Nice: 0

The exact order doesn't matter, but if they cooperate, both can wind up with a score of 16 after 6 turns.

But, I'm not going to be happy with that, either.  Because, Payer One could wind up with 48 points (8 x 6) after 6 turns, so in truth, it would be in both Player One & Player Two's best interest for Player Two to let Player One win in return for some sort of payment.  Or if that's not clear, if they both split Player One's maximum winnings equally, they could both wind up with 24 points.

That of course isn't how the game of Tit for Tat is played, but this exercise was simply to convince you why Tit for Tat proves (suggests, implies, shows, etc) diddly squat in regards to Ultruistic Behavior Being Inevitable.  It doesn't.  Tit for Tat is a math problem.  And if the problem is set up in a different way with different rules, different outcomes and strategies become optimal.

I won't go into these last two as deeply, but before I go, I would like to throw out that there are at least two other arbritary aspects of Tit for Tat that decide the outcome.

The first is that the game is of known length (15 turns, classically, I believe).  And the second is that one is forced to continue to play against the same opponent throughout the competition (whether one wants to or not).  But if the game's length is changed to one of of indeterminite duration (or to as long as both parties wish to interact) and any player may choose to switch who they play with at any time, the optimal strategies may change.  Certainly, playing 'Nice' first will fail against an opponent who plays 'Mean' in a one round competition with the classic payoff chart.  And without repeated interactions, the concept of cooperation or ultruism becomes meaningless.  But by the same token, players who are 'Mean' might find it difficult to find partners to play with if the field is limited.

So, this really isn't an arguement against ultruism (it's benefits or inevitabiltiy in human interaction).  Rather (and much more simply), the game Tit for Tat proves nothing in that regard.  It's simply a math problem.  And it no more proves altruism no more than x+y proves the existence of God.

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copyright © 2013 Brett Paufler
reach me at paufler.net@gmail.com

Chess may (or may not) model medieval tactical warfare.
But is certainly says (or proves) nothing about the relative merits of Knights and Bishops.
In the end, Chess is just a game.
As it Tit for Tat.