Gemara

(Mishnah): If one was on the road when Shabbos was about to come, he may give his wallet to a Nochri [to take it home for him].

Question: Why did Chachamim allow this?

Answer: Chachamim know that a person cannot control himself if he stands to lose much money. If we do not allow him a way to save it, he might carry it four Amos in Reshus ha'Rabim!

(Rava): This is permitted only for his wallet, but not for a Metzi'ah (a found object).

Objection: This is obvious. The Mishnah permits a wallet!

Answer: One might have thought that the same applies to a Metzi'ah. The Mishnah discusses a wallet, for this is the usual case. Rava teaches that this is not so.

Version #1: A Metzi'ah is forbidden only if he did not yet pick it up. If he already picked it up, it is permitted like his wallet;

Version #2 - Question (Rava): What is the law regarding a Metzi'ah that he already picked up?

Since he already picked it up, it is like his wallet;

Or, since he did not toil to acquire it, it is unlike his wallet. (He is not frantic about losing it, so he must leave it there. We are not concerned lest he carry it in Reshus ha'Rabim.)

This question is not resolved.

Bava Metzi'a 9a (Rav Kahana): Riding on an animal in a city does not acquire, for this is abnormal.

Objection (Rav Ashi): If so, one who picks up a wallet on Shabbos does not acquire, for it is abnormal to do so!

That is not so. What he did, he did [and he acquires]. The same applies here!

Avodah Zarah 70a - Isar (the convert): When I was a Nochri, we used to say that Yisraelim do not observe Shabbos. If they did, many wallets would be found in the market [of Yisraelim who were on the road when Shabbos came].

Nochrim do not know R. Yitzchak's law;

(R. Yitzchak): If one found a wallet on Shabbos, he may carry it up to four Amos at a time.

Rishonim

Rambam (Hilchos Shabbos 20:7): One may give his wallet to a [Nochri], Cheresh, lunatic or child, or put on an animal. If none are around, he may carry it home less than four Amos at a time. This is even for a Metzi'ah that came to his hand. If it did not come to his hand, if he can wait there until dark (Motza'ei Shabbos), he does so. If not, he may take it less than four Amos at a time.

Rebuttal (Ra'avad): The last law is totally wrong.

Magid Mishneh: All the Meforshim disagree with the Rambam. They say that just like one may not give to a Nochri a Metzi'ah that did not yet come to his hand, he may not take it less than four Amos at a time. The Rambam was asked about this, and answered (Pe'er ha'Dor 35) 'the Gemara does not distinguish between a wallet and a Metzi'ah regarding going less than four Amos at a time, for this is very covert, like one who found Tefilin [and wears them until he reaches a guarded place]. He does not need anything or anyone else to help him. No one will know, and he will not be frantic. Therefore, we are not concerned lest he carry four Amos [at once] in Reshus ha'Rabim. This is my opinion. One who wants to argue, he will argue.' The Rambam compares this to one who found Tefilin. Therefore, if he can wait there until dark, he does so. One may not take a Metzi'ah if he could wait until dark. This is permitted only for Tefilin. If not, he may take it less than four Amos at a time. The Rambam himself admitted that his opinion is not necessarily true. Therefore, one should be stringent like the other Meforshim.

Rosh (Bava Metzi'a 1:22): Rav Ashi said that it is abnormal to pick up a wallet on Shabbos. This shows that if one found a wallet on Shabbos, he may not pick it up, even though he fears lest someone else take it. If it were permitted, how do we compare this to riding in a city, which is abnormal even regarding a Metzi'ah, for one can lead it? It would be normal to pick up a wallet! Rather, this shows that it is forbidden. In Shabbos, we say that one's wallet is permitted, but not a Metzi'ah. This is not only regarding the Heter to give it to a Nochri or carry it less than four Amos at a time. Rather, one may not even pick it up, for it is Muktzeh.

Poskim

Shulchan Aruch (OC 266:1): If one found a Metzi'ah [on Shabbos], he may not give it to a Nochri [to take it home for the Yisrael] unless he picked it up before Shabbos, for then it is like his wallet.

Shulchan Aruch (7): One may give [to a Nochri, Cheresh, lunatic or child, or put on an animal] his wallet, or a Metzi'ah that came to his hand.

Mishnah Berurah (19): It is permitted only if it came to his hand before Shabbos.

Shulchan Aruch (ibid.): If it did not come to his hand [before Shabbos], he may not.

Beis Yosef (DH Kasav ha'Rambam): Also the Ran agreed with the Ra'avad, unlike the Rambam.

Gra: Migdal Oz and the Ramban defended the Rambam. He learns from Avodah Zarah 70a. The other Poskim must say that there, 'ha'Motzi' refers to one who took out his wallet b'Shogeg. The Rambam explains simply, that he found it on Shabbos. The Gemara connotes like this. It says 'many wallets should be found in the market. This is because we left the law at the Torah level. There is no difference between one's wallet and a Metzi'ah. However, Bava Metzi'a 9a connotes like the other Poskim.

Bi'ur Halachah (DH Aval): The Pri Megadim forbids less than four Amos at a time through one person, but permits from person to person, and each carries less than four Amos. It says so in 349:3 and the Taz there (1). Chachamim did not decree about less than four Amos at a time in such a case. This is only if one found something that one may move. If he found Muktzeh, it is forbidden, like in Sa'if 13.

Kaf ha'Chayim (37): Nishmas Adam (54:3) says that if the person will not accept the Isur, and he will come to bury it in the ground, perhaps one may rely on the Rambam to take it less than four Amos at a time.

Shulchan Aruch (13): If one found a wallet on Shabbos, one may not move it, even though he fears lest someone else take it.

Beis Yosef (DH Kasvu): The Rosh, Ran in the name of Tosfos, Hagahos Maimoniyos (2:6) and R. Yerucham forbid picking up a found wallet on Shabbos.

Magen Avraham (21): One may not tell a Nochri to take it, like in Sa'if 1.

Eliyahu Rabah (26): Here is different. Sa'if 1 discusses when the Yisrael already transgressed and picked it up, or it was something that one may move. Therefore, one may not [command a Nochri]. Here, the Yisrael did not acquire it yet. The Nochri could take it for himself. Therefore, it is like telling the Nochri to take the Nochri's item. If the Nochri gives it to the Yisrael, this is like a mere gift. Also, the Rambam totally permits.

Kaf ha'Chayim (65): Tosefes Shabbos disagrees with Eliyahu Rabah, but permits for a great need.

Gra: We learn from Bava Metzi'a 9a. This is unlike the Rambam. See Sa'if 7.

Mishnah Berurah (38): One may not take it, even if he will stay there and not move it four Amos, due to Muktzeh. We do not permit due to a loss, since he did not acquire it yet. The Magen Avraham forbids telling a Nochri to take it. The Pri Megadim brings Eliyahu Rabah, who permits, and also R. Akiva Eiger is lenient to tell a Nochri to pick it up, but not to take it to the house. If the Nochri does so by himself, all agree that one need not stop him.

Kaf ha'Chayim (66,67) Tosefes Shabbos forbids in a Chatzer, for he can sit there and guard it. One may guard Muktzeh on Shabbos. All the more so one may tell a Nochri to guard it!

Bi'ur Halachah (DH Matza): If one transgressed and picked it up, he must carry it less than four Amos at a time, and at the end he casts it in back of himself, so there will not be Hanachah.

Bi'ur Halachah (DH Asur): If mostly Yisre'elim live in the area, and he picks it up in order to return it to the owner later, surely we do not permit carrying less than four Amos at a time, for he is not frantic. It is not clear whether we permit Muktzeh for this. See 586:22 and the Gra there. (The Shulchan Aruch permits blowing a Shofar that a Nochri brought from outside the Techum, or that he made on Yom Tov. The Gra brings that Hagahos Ashri permits only because it is not Muktzeh, but the Mordechai permits even Muktzeh.)

Bi'ur Halachah (DH Pen): One may be lenient to move it with his foot.