# LDT (and everything else) can be irrational

> you should not reject the 'offer' of a field that yields an 'unfair' amount of grain! - [*Ultimatum Game (Arbital)*](https://arbital.com/p/ultimatum_game/)

In this post, I demonstrate a problem in which there is an agent that outperforms [Logical Decision Theory](https://arbital.com/p/logical_dt/), and show how for any agent you can construct a problem and competing agent that outperforms it. Defining rationality as winning, this means that no agent is rational in every problem.

Symmetrical Ultimatum Game
==========================

We consider a slight variation on the ultimatum game to make it completely symmetrical. The symmetrical ultimatum game is a two-player game in which each players says how much money they want. The amount is a positive integer number of dollars. If the sum is ≤$10, both players get the amount of money they choose. Otherwise, they both get nothing.

Now consider the decision problem of playing the symmetrical ultimatum game against a logical decision theorist.

A casual decision theorist does particularly poorly in this problem, since the LDT agent always chooses $9 leaving the casual decision theorist with $1.

How does a LDT agent fare? Well, logical decision theory is still a bit underspecified. However, notice that this question reduces to "how does a LDT agent do against a LDT agent in a symmetrical game?". Without knowing any details about LDT, we must conclude that the expected value is at most $5.

What about a rock with $9 painted on it? The LDT agent in the problem reasons that the best action is to choose $1, so the rock gets $9. 

Thus, $9 rock is more rational than LDT in this problem. □

You can't become $9 rock
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Now, what makes this problem particularly difficult is how picky the LDT agent in the problem is. If based on the previous you decide to "become $9 rock", the LDT agent will defect against you. If based on the previous section you build a robot that always chooses $9, the LDT agent will defect against that robot. Only a truly natural $9 rock can win.

No agent is rational in every problem
=====================================

Consider an agent X. There are two cases:

1.  Against $9 rock, X always chooses $1. Consider the problem "symmetrical ultimatum game against X". By symmetry, X on average can get at most $5. But $9 rock always gets $9. So $9 rock is more rational than X.
2.  Against $9 rock, X sometimes chooses more than $1 (thus getting nothing). Consider the problem "symmetrical ultimatum game against $9 rock". X on average gets less than $1. But an agent that always picks $1 (that is, a $1 rock) always gets $1. So $1 rock is more rational than X.

□

Implications
============

I still have an intuition that LDT is the "best" decision theory so far. See   
[*Integrity for consequentialists*](https://forum.effectivealtruism.org/posts/CfcvPBY9hdsenMHCr/integrity-for-consequentialists-1) for practical benefits of a LDT style of decision making.

However, there can be no theorem that LDT is *always* rational, since it isn't. And replacing LDT with a different agent can not fix the problem. Notice that, as a special case, humans can never be rational.

This seems to suggest some sort of reformulation of rationality is needed. For example, given LDT's *reasonableness*, one option is to violate the thesis of [*Newcomb's Problem and Regret of Rationality*](https://www.lesswrong.com/posts/6ddcsdA2c2XpNpE5x/newcomb-s-problem-and-regret-of-rationality) and simply *define* rationality to be LDT.