There is two tech. leaders in a team (for example Solution Architect and Team Leader or two Senior Devs.) They have a dispute about a technical approach to use for an issue resolution. They can’t come up to a solution, concession of each side will be considered as a weakness.
What will you do in this situation?
While I agree with @DavidEspina (as usual), it might be useful to explore alternatives.
You hint that the underlying cause is a conflict of personalities/egos ("showing weakness") I abstain on the resolution of these conflicts, except to say that they're going to be messy and unpleasant.
But another possible source of conflict might be an unvoiced, underlying disagreement. My first step would be to listen to both of the experts, but I would be tempted to ask them both to drop the question on the table, and instead to ask:
If a third party were to suggest another approach - one that is not related to either of your approaches - how should we measure that approach? How would we judge if it was successful? How would we estimate the cost & benefits?
There is a chance that the disagreement is not the one over which they are arguing, but a more fundamental disagreement about what "successful" means, or about which feature to optimize.
You implement a formal decision making process: you bring in their superior, you bring in a a handful of other technical experts, you give each party a period of time, say 30 minutes, to argue their case, you allow the other technical experts to deliberate and either vote or establish a consensus, and you get their superior to uphold the decision made by the vote. There are other formal methods you can use but the idea behind this is to remove the emotional aspects to the argument and the perceived threat to one's ego.
You have different issues depending on the roles of the two senior people. If it is Solutions Architect v Team Leader, the roles should be defined such that responsibility for design-related issues should lie with the Architect, while anything to do with the implementation of the design should lie with the Team Leader. They should be prepared to accept that.
It's a bit harder with two Senior Developers, especially if they have the same level of technical authority, so in this case either the PM or the line manager of the two needs to make a decision. I might try to suggest that whoever's solution is chosen, that person needs to be 100% accountable for making the solution work, and so if they have any doubts about their solutions, they should speak up now. Then make a decision and select one of them to take ownership (it doesn't matter which one) - and ask the other to take 100% ownership of some other part of the project to free up the first one to fix the issue.
Either way it isn't easy but you need to find a way to solve the problem otherwise it will block the whole project. And just one further thought... If you have a technical responsible person (as per a project organised using Prince 2 methodology), you could bring that person into the loop as the ultimate arbiter of anything technical.
David & Lain already mentioned official approaches to involve external experts. I just have another thought on this: allowing the team to make the decision (Ex: voting to select 1 approach from the 2).
This will help to build up a self-organization mindset & a safe environment for constructive disagreement or even failure.
I see absolutely no reason here to conclude "weakness," nor particularly to conclude that the two of them "can't come up with" an equitable solution.
