A seller is holding a sealed$-$bid, second$-$price auction for a rare case of wine. A group of II individuals is interested in bidding for this wine, with each bidder valuing the wine only for personal consumption and not for resale. The bidders$$ values for the wine, vivi$,$ are independent and private. You are one of the bidders, labeled bidder ii$,$ and your value for the wine is vivi$.$

You are asked to determine how you would bid in each of the following situations. Provide a brief explanation for your decision$$formal proofs are not required.

$($a$)$ You are aware that a subset of the bidders will collude. This collusive group will nominate one member to submit a "real bid" equal to vv$,$ while all others in the group submit bids of $00.$ You are not part of this collusive group and cannot communicate or coordinate with any other bidder.

b$)$ You and the other bidders have been informed of a significant change to the auction process. Although the seller will collect bids as usual, the auction will no longer follow the standard second$-$price auction rules. Instead, after collecting the bids, the seller will inform everyone that a fictional bidder submitted the highest bid and "won" the auction. This fictional bidder does not actually exist, so the seller retains the item after the auction concludes.

Subsequently, the seller will privately contact the highest actual bidder to inform them that the fictional bidder defaulted on their purchase. The seller will then offer the highest actual bidder the opportunity to buy the item at the exact price they bid in the auction.

No collusion with other bidders is permitted. The question is whether this new setup would cause you to bid differently than your true valuation, and if so$,$ in which direction your bid might change. Note that you are not required to derive the optimal bidding strategy but rather provide a rationale for how this change affects your bidding behavior.