I have a very general question to ask - if I am getting multiple equilibrium and I have to check which one will be picked ie. which Solution is stable, can I check it by comparing whichever produces maximum payoff for the principal. Basically, is stability same as absolute maximum point for the principal. Or is it the case that profit maximisation and stability are two different concepts
Stability is a very broad subject, and in general, has nothing to do with profit maximization.
Obviously, the results vary from model to model, and it is possible that in a specific model profit maximization and stability could be linked.
But stability is a very different concept from profit maximization, also because profit maximization is usually, as it is studied in standard microeconomics, a static concept, whereas stability is a notion that pertains to dynamical systems, that is, to speak of stability we must have models in which time enters in an essential way.
Therefore, to speak of stability of an equilibrium we must have dynamical models formulated as dynamical systems, which are usually described by discrete dynamical systems (time is a discrete variable) or by differential equations (time is a continuous variable), so we speak of discrete or continuous dynamical systems.
Dynamical systems is a theory derived from classical mechanics, and so are the concepts of equilibrium and stability of equilibrium. $^1$
For a mechanical system an equilibrium point or fixed point or null solution is defined as a configuration such that the system, if initially in that configuration, remains there. Formally, a fixed point or equilibrium is a point where the change of the variables under consideration, that is their derivatives with respect to time in continuous systems or their difference between two subsequent periods of time in discrete systems, are zero.
An equilibrium is said stable if the system, initially at the equilibrium point, and subject to a small perturbation, tends to come back to the equilibrium point, or better
A system is stable if, when perturbed slightly from its equilibrium state, all subsequent motion remains in a correspondingly small neighborhood of the equilibrium. $^2$
Instead, the equilibrium is said unstable if, after a perturbation, the system tends to go away from equilibrium.
Therefore, in dynamical systems there is a classification of equilibrium points, for instance equilibrium points are called sources (or repellers, or unstable) or sinks (attractors or stable).$^3$
Moreover, we must remember that, in the case of multiple equilibria, there can be more than one stable equilibrium, so that speaking of the stable equilibrium is not correct.
Stability and profit maximization are different and independent concepts
Coming back to economics, a static model per se cannot say nothing about stability of equilibria, but to speak of stable or unstable equilibria in connection with a static model we must attach to it a dynamical system, making assumption about the evolution of the variables in time, which cannot be present in the static model.
As the subject of stability is very complex, I can just give some illustrative examples to give an idea of the concepts I mentioned: the distinction between statics and dynamics, the irrelevance of profit maximization in determining the stability of a system, and the possibility of existence of more than one stable equilibrium.
I consider first of all the well-known model of market equilibrium in perfect competition.
The traditional model of competitive equilibrium in a market is usually formulated in static terms, as equilibrium between demand and supply in that market, and can be represented through the usual graph:
where, as usual, $P^e$ and $Q^e$ are the equilibrium price and quantity, and equilibrium is defined as equality between demand and supply.
Where is profit maximization here? The supply curve, in a competitive market, is derived from profit maximization of the firms, so that each point of the supply curve represents a profit maximizing quantity, depending on the market price. Of course, also in equilibrium the representative firm is maximizing its profit.
Is the equilibrium represented in the graph stable or instable? The model cannot answer this question, because it is a static model, which says nothing about the behavior of the variables outside equilibrium, in particular if they will reach or not the equilibrium point.
To establish if and under which assumptions the equilibrium is stable, we must attach to the static model a dynamical model, which describes the behavior of the variables in time, if they are away from equilibrium.
That dynamic model can be formulated in different ways, and under different assumptions, so that the same static model represented in the graph can exhibit a stable or unstable equilibrium according to different assumptions.
A classical way to attach an underlying dynamics to the supply-demand models, is to hypothesize a so-called cobweb dynamics$^4$, under which the price and the quantity vary according to the expectations of the agents about future prices.
The result of a cobweb dynamic is that the market equilibrium will be stable or unstable according to the parameters of the model, in particular the slopes of the demand and supply functions. The graphs below represent two cases of, respectively, stable and unstable cobweb process: in the first case the system reaches equilibrium in time, in the second it goes far away from equilibrium, as time passes:
Of course, other dynamic assumptions can be made about the model, for instance a dynamic based on excess supply and demand, and the stability or not of the equilibrium must be established on the base of different assumptions.
And we see that, as in equilibrium the representative firm always maximizes profits, as on all the supply curve, this maximization has nothing to do with stability or instability.
In case of multiple equilibria, there can be many stable equilibria
There are true dynamic models, that is models that are dynamic from the start, but also in that case nothing implies that there will be a unique stable equilibrium, if equilibria exists.
A well-known example is the Solow growth model, which in its standard formulation has only one stable equilibrium, but in some of its versions there can be cases of multiple equilibria, and more than an equilibrium can be stable. See the following graph:
In the picture above the saving assumptions are such that the saving curve has a 'waved' form, and intersects the line $(n+\delta)k$ at three points: there are three equilibrium points, and two of them, $A$ and $C$, are stable, instead $B$ is unstable, as indicated by the arrows.
Stability of equilibrium has nothing to do with efficiency
Moreover, we must recall that the concept of equilibrium has nothing to do with efficiency, as you seem to suggest in your comment above.
An equilibrium, by definition, in a dynamic system, is just a fixed point, that is a point where the dynamics of the variables under consideration stops, and the system is in a 'state of rest'.
Nothing ensures that this state is efficient or even desirable. An equilibrium state can be also a 'bad' equilibrium: think of an equilibrium of unemployment in keynesian theory or again of some case of modified Solow Growth model, where in some circumstances what is called the poverty trap $^5$ can arise, namely a situation in which the system is stuck at very low level of income and capital accumulation.
The point of equilibrium $A$ in $Fig. 4$ above can represent a poverty trap: the economic system, let alone, without an outside intervention, for instance of the Government, remains indefinitely in a state of very low standard of life. An economic policy is needed to increase income and capital, for instance a so-called big push, a massive intervention, to take the capital per capita beyond point $B$. $$***$$
The distinction between stable and unstable equilibrium points is not exhaustive, there can be other distinctions, as asymptotic stability, neutral equilibria, local or global stability.
For a discussion of stability in economic dynamics, and also of different types of stability, see Gandolfo G., Economic Dynamics, Springer, 2009.
$^1$ It must be recalled that here we are speaking of equilibria in dynamical systems, which is the mathematical environment where we can speak of stability of equilibrium. We are not speaking of the concept of equilibrium in economics in general, where the concept can be different, for instance equality between demand and supply.
$^2$ Gandolfo G., Economic Dynamics, Springer, 2009, p. 353.
$^3$ For an extensive discussion of these classifications and of stability see Strogatz Stephen H., Non Linear Dynamics and Chaos, CRC Press, 2015.
$^4$ Nerlove, M., (1958), Adaptive expectations and Cobweb Phenomena.
$^5$ See also, for example, http://growth-institutions.ec.unipi.it/pages/Human_Capital/education_poverty.pdf