In order for a four way tie to occur each player would have to have a 1st, 2nd, 3rd, and 4th place finish. No other result in four games will give 7 points. Since each player has to have one of each place the results of the first game aren’t important to the calculations. To make the explanation a bit simpler let’s assume that in game one P1 was 1st, P2 was 2nd, P3 was 3rd, and P4 was 4th.
The probability of a 4-way tie is simply the number of outcomes that result 4-way tie divided by the total number of possible outcomes. Each match has 24 possible outcomes. For the three remaining matches there are 24^3=13824 possible outcomes. P1 needs a 2nd, 3rd, and a 4th in these matches. There are six ways that this can occur. The next step is to figure out how many possible ways there are for this to result in all players tying.
Let’s assume that P1 finished in 2nd in game two, 3rd in game three, and 4th in game four. That leaves three possibilities for how P2 can finish (341, 413, 143). If P2 finishes 341, then P3 has to finish 412 and P4 has to finish 123. P2 finishing 413 also leaves only one way P3 and P4 can finish (P3:142, P4:321). If P2 finishes 143, then there are two different ways that P3 and P4 can finish (P3: 412, P4: 321 or P3:421, P4:312). This gives us four possible ways for there to be a 4-way tie for players 2-4 for each of the original six possibilities of how P1 finished. The result is there are 4*6=24 ways for there to be a 4-way ties out of 24^3 total possibilities for the final three games. This gives us a 1/576 probability of a 4-way tie.