Vickrey Auctions

Table 1: Payoff matrix
  1. (v_1, v_1, v_3, ……v_n) : Consider this bidding profile. Player 1 gets the object in this case according to our rules of tie-breaking. He obtains a payoff of zero with this bid. With a higher bid, he would still incur a payoff of zero as the second-highest bid is equal to his valuation v_1 and with a lower bid, he would lose the object to player 2. So player 1 has no incentive to deviate. Similarly, player 2 currently has a payoff of zero. With a higher bid, he would acquire the object, but incur a negative payoff of (v_2 — v_1). With a lower bid, he would still have a payoff of zero. So player 2 also has no incentive to deviate. Readers can easily verify for other players as a friendly exercise.
  2. (v_2, v_2, v_3,…….v_n) : This bidding profile is another NE where player 1 wins the prize. Readers can verify why this is a Nash equilibrium by following a similar train of thought as in the last case.
  1. Vickrey, W.(1961) “Counterspeculation, auctions, and competitive sealed tenders,” Journal of Finance, 16, 8–37.
  2. https://en.wikipedia.org/wiki/William_Vickrey

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