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Suppose there are 4 individuals and two items in this particular(?) auction. I understand that the dominant strategy in second price auction is to bid your true value if there is a single item but I'm having a hard time understanding why that would change at all if there were two items.

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    $\begingroup$ Good question but what precisely is the auction structure? $\endgroup$ – user17900 Sep 27 '19 at 11:10
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This question is hard to answer, because it is not clear what the appropriate multi-unit version of the single-unit second-price auction (SPA) actually is and which economic environment you have in mind. There are several possibilities, and I mention some.

There are 4 bidders and two units.

  1. All bidders draw a two values $v_{1i},v_{2i}$ representing the willingness-to-pay for unit $k\in\{1,2\}$, having both units is valued $v_{1i}+v_{2i}$ (only one draw such that $v_{1i}=v_{2i}$ is also possible). There are two separate SPAs and in each one every bidder truthfully bids their value in equilibrium.

  2. All bidders draw two values $v_{1i}\geq v_{2i}$ representing the willingness-to-pay, where having one unit is valued $v_{1i}$, having both units is valued $v_{1i}+v_{2i}$. There is one uniform price auction, i.e., all bidders submit a bid vector $(b_1,b_2)$, the highest two bids win and the payment per unit is the highest losing bid. It is a dominant strategy to bid truthfully on the first unit $b_1=v_{1i}$, but not necessarily for the second unit, because the second bid could determine the price which a winner of one unit must pay.

  3. All bidders draw two values $v_{1i}\geq v_{2i}$ representing the willingness-to-pay, where having one unit is valued $v_{1i}$, having both units is valued $v_{1i}+v_{2i}$. There is one Vickrey auction, i.e., all bidders submit a bid vector $(b_1,b_2)$, the highest two bids win and the payments are the highest losing bids of the opponents. It is a dominant strategy to bid truthfully on both units with the same argument as with a single unit.

  4. All bidders have single-unit demand and draw a single value $v_i$. Two SPAs are executed sequentially. The three remaining bidders bid truthfully in the final auction, but none of the four bidders in the first auction do, because there is a positive option value of losing the first auction: participating in the second auction.

  5. All bidders have single-unit demand and draw a single value $v_i$. A generalized second-price auction is executed, i.e., the highest bidder wins and pays the second-highest bid, the second-highest bidder wins and pays the third-highest bid. If two bidders submit high bids and the other two submit low bids, the highest bidder would prefer to bid lower in order to win the second unit instead to then pay the third-highest and lower bid.

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