A one-way rivalry is a rivalry where one of the participants is much more emotionally invested in the conflict. This state could arise if one of the parties (A) to the rivalry objectively outmatches the other (B) in significant way. The inferior party (B) will continue to value the prospect of winning, while (A) will shift focus toward other competitors that provide more of a challenge. Alternatively, (A) may simply have lost interest in the rivalry.
(B) may perceive their opponent (A) to be elitist or condescending if the one-way status of rivalry becomes clear. Conversely, (A) may consider (B) to be desperate or insecure.
1. In sports, the Mets-Yankees rivalry is a great example of a one-way rivalry. By virtue of their geographic proximity, some conflict between the teams is inevitable. However, the Yankees are widely acknowledged to be the more powerful of the two and find the Red Sox to be a more worthy opponent (though the Sox-Yankees has had elements of a one-way rivalry in the past).
2. Perhaps the most significant example in US society is the South's antagonism for the North. Many Southerners still bear great ill will toward the North for their victory in the Civil War and subsequent control during Reconstruction, some even believing that the North may be actively attempting to "keep them down". Actual antagonism from Northerners toward the South is generally limited to jokes about the Southern accent and the like.