Let's continue with bargaining problems and consider other classical solutions, other classical bargaining solutions. The first one we'll consider is the Kalai-Smorodinsky solution for two-player game. We will only consider a case of two player game because in classical literature it is defined only for two-player game. Okay, so the Kalai-Smorodinsky solution is the maximum point of the bargaining set on the interval connecting the disagreement point and point v1(S) and v2(S). Kalai-Smorodinsky solution also satisfies a number of axioms. It is proved that the solution satisfies axioms one, two, three and five if and only if it is the Kalai-Smorodinsky solution. So, the first, second and third axioms is the same axioms that we use for Nash bargaining solution, but the last one is called the monotonicity. So, we choose, instead of axiom four, we use the monotonicity axiom. It says that, If we consider the initial bargaining set and calculate the corresponding bargaining solution, for example, the Kalai-Smorodinsky solution for this bargaining set. Then we will consider the subset or any subset of the bargaining set but with the same border points v1(S) and v2(S), then the bargaining solution defined for this subset will be less or equal to the bargaining solution defined on the initial bargaining set. The physical meaning of that is if we reduce the number of outcomes or reduce number of possible payoffs, then of course, the solution will also be less or equal than the solution of the initial game. The next classical solution is called the Egalitarian solution, or it's also called the Egalitarian rule in other fields of game theory. This one is a very practical solution and it is very easy to calculate. So, the egalitarian solution is the maximum point of the bargaining set with equal coordinates, how it can be constructed. Let's look on the figure which is shown on this slide. And then here, what we can do is we can construct line as it is done on the slide and then we can pick the point which is located on the intersection of the Pareto optimal border or the border of the bargaining set and this line, then this point will be the Egalitarian solution. As it turns out, the Egalitarian solution also satisfies three axioms. The first one is called the Weak Pareto optimality, the second one is the symmetry which we already considered and the last one is the strong monotonicity. Okay, so the following theorem can be proved. Solution satisfies the axioms one, two and five if and only if it is the Egalitarian solution. So, what is the strong monotonicity? Monotonicity property means that if we take the subset of the initial bargaining set with equal border points, then the following conditions hold. But in here we say that for any subset of the initial bargaining set, the bargaining solution for the initial bargaining set is less or equal to the bargaining solution of the bargaining subset. Let's go back to the example of Battle of Sexes and let's calculate or let's define the solutions for this game model. As you can see, Nash bargaining solution, Kalai-Smorodinsky solution and Egalitarian solution for this specific game model are equal. Why is that so? It is only because the Pareto optimal set of outcomes which is the interval a-b is symmetrical to the disagreement point d. Well, let's consider another example where the result will not be so obvious. So, let's consider the example which is based on the bimatrix game where each player has three pure strategies. For this example we can also calculate the Nash equilibrium. The payoffs in the Nash equilibrium are eight and five. We can use it as a disagreement point. But let's suppose that disagreement point for this particular game model is vector three and three. So, we suppose that the players, they agree on cooperation if their payoffs would be more than three for both of the players. Then, what we can do is we can again construct the set of the bargaining set for that using the disagreement point and we can calculate the corresponding Nash bargaining solution, Kalai-Smorodinsky solution and Egalitarian solution. For this particular example, we can see that the solutions are different and we can study them. On this slide you can see a list of references where you can study of how the theorems corresponding to the Kalai-Smorodinsky solution and Egalitarian solutions are proved and you can find more examples on this topic.