From Middle English over hond, from Old English ofer- + hand (superior control; superior position). Not, as supposed, from a card game or counting-out game.

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the upper hand (usually uncountable, plural upper hands)

1. (idiomatic) Advantage or control.
   • 1855, Washington Irving, Guests from Gibbet Island:

     There was no refusing him, for he had got the complete upper hand of the community, and the peaceful burghers all stood in awe of him.

   • 1883, Robert Louis Stevenson, chapter 23, in Treasure Island:

     [C]uriosity began to get the upper hand, and I determined I should have one look through the cabin window.

   • 1911, Joseph Conrad, chapter 1, in Under Western Eyes:

     There it was Razumov who had the upper hand, in a composed sense of his own superiority.

   • 2003 February 14, Christine Gorman, “Playing Chicken With Our Antibiotics”, in Time:

     And because they live everywhere and reproduce quickly, bacteria have the upper hand.

   • 2020 August 26, Andrew Mourant, “Reinforced against future flooding”, in Rail, page 61:

     "We've now protected the line from similar-sized flooding-events and bigger ones," he says. That's quite some claim for a line where floods have often had the upper hand in the past 16 years, causing track bed and embankments to be rebuilt.

2. (obsolete) The place of honour accorded to a social superior when walking together; the right of way in walking