The process of pair partnership formation is an important infrastructure for many plausible MAS applications. Each agent evaluates potential partner agents, where each potential match yields a different utility. Commonly, the utility associated with a given agent partner in such two-sided search processes may change over time. This change in the agent's future attractiveness to potential partners significantly increases the complexity of the agent's decision making process regarding the set of agents it is willing to partner with. In this paper we analyze the special dynamics and present equilibrium characteristics of such a model. The agents can gain a utility derived from the partner agent's type. However, as an agent has an incentive to extend its search for a better type partner, the benefit that can be offered to potential partners reduces as the search proceeds. We introduce a two-sided model which takes into consideration a continuous decrease in the agent's type and formulate the appropriate equilibrium equations. The suggested equilibrium analysis yields an algorithm for the calculation of the agents' equilibrium strategy. Special emphasis is placed on the scenario where an agent's attractiveness is influenced by an additional dimension other than just time. Simulation results are presented to illustrate the findings.