The argument is based on the famous prisoner dilemma result. For example, consider the following situation. We have firm A and firm B. If they both collude and set high price (high P) they both can get profit $\\\$7$, If A sets high price but B undercuts A (note undercutting is not the same as engaging in dumping see end of answer) then A will earn 0 profit because everyone will want to buy goods from B and B earns $\\\$10$ profit. If A chooses low P and B high P the opposite will happen. If they both choose low price they both earn $\\\$5$ profit. Moreover, we will assume that they cannot enter into a contract that would allow them to make commitment to high prices that would be enforceable in law.
The situation can be visualized in the matrix below:
High P Low P
Firm A High P $7,$7 $0, $10
Low P $10,$0 $5, $5
What will happen in this case? Lets start looking at the problem from perspective of firm B:
- what happens if firm A chooses high price? You can easily see that the most rational response for firm B is to chose low price - thats how they maximize their profit as getting profit of $\\\$10$ is better than profit of $\\\$7$.
- What happens if firm A chooses low price? In this case again it would be most rational for firm B to choose low price because its better to get at least the $\\\$5$ profit than zero.
Thus setting low price is strictly dominant strategy for firm B and if firm B is rational it should always set prices low. The mirror argument applies to firm A as well. Hence, due to this collusion cannot be sustained.
The simple prisoner dilemma model captures the core of the argument but also its not entirely realistic as in real life firms can be argued to 'play' repeated game of competition and in repeated games conclusions can be sometimes sustained depending situation. For example, if firm play the above game repeatedly they can collude because repeated games allow for richer amount of strategies such as trigger strategies where firms can try to punish each other for deviating from the collusion equilibrium.
You can learn about these more complex situations in some industrial organization textbooks such as Industrial Organization: Markets and Strategies by Martin Peitz and Paul Belleflamme.
Nonetheless, in real life also most countries set very high fines and penalties for colluding firms as well as rewarding any firm that 'snitches' on other colluding firms. Thanks to the fact that these penalties are very strict it is incredibly hard for firms to sustain any explicit collusion which is why they are rare but still occur sometimes - see some EU cases here (although tacit collusion - i.e. collusion where firms set high prices without actually making some secret agreement can still occur).
Furthermore, note selling for price lower than your competitor is not called dumpling. Dumping means selling products for less than it cost to produce them not less than others. Dumping is a separate issue altogether.